Trace, Metric, and Reality: Notes on Abstract Linear Algebra

Adam Coffman

Department of Mathematical Sciences Purdue University Fort Wayne 2101 E. Coliseum Blvd. Fort Wayne, IN 46805-1499

   Email address: CoffmanA@pfw.edu
   URL: http://pfw.edu/math/

2020 Mathematics Subject Classification. Primary 15-02 Secondary 15A15, 15A63, 15A69, 18M10, 53B30, 53C15

Key words and phrases. Canonical map, linear algebra, trace, indefinite metric, tensor, vector valued form, complex structure operator.

This project was supported in part by:

A sabbatical semester at Purdue University Fort Wayne in 2020,

A sabbatical semester at Indiana University - Purdue University Fort Wayne in 2015, A sabbatical semester at Indiana University - Purdue University Fort Wayne in 2005,

A Summer Faculty Research Grant from Indiana University - Purdue University Fort Wayne in 2001, A Summer Faculty Research Grant from the Purdue Research Foundation in 1998.

Abstract. Elementary properties of the trace operator, and of some natural vector valued generalizations, are given basis-free statements and proofs, using canonical maps from abstract linear algebra. Properties of tensor contraction with respect to a non-degenerate (but possibly indefinite) metric are similarly analyzed.

Contents

Preface v

Overview 1

Chapter 0. Review of Elementary Linear Algebra 3 0.1. Vector spaces 3 0.2. Subspaces 5 0.3. Additive functions and linear functions 7

Chapter 1. Abstract Linear Algebra 11 1.1. Spaces of linear maps 11 1.2. Tensor products 15 1.3. Direct sums 25 1.4. Idempotents and involutions 32

Chapter 2. A Survey of Trace Elements 47 2.1. Endomorphisms: the scalar valued trace 47 2.2. The generalized trace 53 2.2.1. Defining the generalized trace 53 2.2.2. Properties of the generalized trace 55 2.3. Vector valued trace 70 2.3.1. Defining the vector valued trace 70 2.3.2. Properties of the vector valued trace 73 2.4. Equivalence of alternative definitions 77 2.4.1. A vector valued canonical evaluation 78 2.4.2. Coevaluation and dualizability 92

Chapter 3. Bilinear Forms 105 3.1. Symmetric bilinear forms 105 3.2. Metrics 109 3.3. Isometries 110 3.4. Trace with respect to a metric 113 3.5. The induced metric on Hom(U, V) 115 3.6. Orthogonal direct sums 117 3.7. Topics and applications 119 3.7.1. Foundations of geometry 120 3.7.2. More isometries 122 3.7.3. Antisymmetric forms and symplectic forms 126 3.7.4. More direct sums 127 3.7.5. Isotropic maps and graphs 132 3.7.6. The adjoint 133 3.7.7. Some formulas from applied mathematics 136 3.7.8. Eigenvalues 142 3.7.9. Canonical metrics 143

Chapter 4. Vector Valued Bilinear Forms 147 4.1. Transpose for vector valued forms 147 4.2. Symmetric bilinear forms 159 4.3. Vector valued trace with respect to a metric 163 4.4. Revisiting the generalized trace 169 4.5. Topics and applications 174 4.5.1. Quadratic forms 174 4.5.2. Algebras 178 4.5.3. Curvature tensors 182 4.5.4. Partially symmetric forms 185

Chapter 5. Complex Structures 189 5.1. Complex Structure Operators 189 5.2. Complex linear and antilinear maps 192 5.3. Commuting Complex Structure Operators 197 5.3.1. Two Commuting Complex Structure Operators 197 5.3.2. Three Commuting Complex Structure Operators 205 5.3.3. Four Commuting Complex Structure Operators 217 5.4. Real trace with complex vector values 229

Chapter 6. Appendices 233 6.1. Appendix: Functions and Binary Operations 233 6.2. Appendix: Quotient spaces 236 6.3. Appendix: Construction of the tensor product 237 6.4. Appendix: Comments on [C_{2}] 239

Bibliography 241

Preface

These notes are a mostly self-contained collection of some theorems of linear algebra that arise in geometry, particularly results about the trace and bilinear forms. Many results are stated with complete proofs, the main method of proof being the use of canonical maps from abstract linear algebra.

So, the content of these notes is highly dependent on the notation for these maps developed in Chapter 1. This notation will be used in all the subsequent Chapters, which appear in a logical order, but for 1 < m < n, it is possible to follow Chapter 1 immediately by Chapter n, with only a few citations of Chapter m. To review the elementary prerequisites, some foundational material appears in Chapter 0 and the Appendices.

In such a collection of results, there will be several statements which will not be needed in later Lemmas, Theorems, or Examples, and can be skipped without losing any logical steps. Such statements will be labeled “Proposition” or “Exercise,” with a short proof following from a “Hint” or left to the reader entirely. There are a few statements which are needed in later steps but whose proofs do not fit the basis-free theme; they are labeled “Claim,” with proofs left to the references.

Adam Coffman

Overview

The goal of these notes is to present the subject of linear algebra in a way that is both natural as its own area of mathematics, and applicable, particularly to geometry. The unifying theme is the \underline{trace} operator on spaces of linear maps, and generalizations of the trace, including vector valued traces, and traces with respect to non-degenerate inner products. The emphasis is on the canonical nature of the objects and maps, and on the basis-free methods of proof. The first definition of the trace (Definition 2.3) is essentially the “conceptual” approach of Mac Lane-Birkhoff ([MB] §IX.10) and Bourbaki ([B]). This approach also is taken in disciplines using linear algebra as a tool, for example, representation theory and mathematical physics ([FH] §13.1, [Geroch] Chapter 14, [K]). Some of the subsequent formulas for the trace (Theorem 2.10, and in Section 2.4) could be used as alternate but equivalent definitions. In most cases, it is not difficult to translate the results into the usual statements about matrices and tensors, and in some cases, the proofs are more economical than choosing a basis and using matrices. In particular, no unexpected deviations from matrix theory arise.

Part of the motivation for this approach is a study of vector valued Hermitian forms, with respect to abstractly defined complex and real structures. The conjugate linear nature of these objects necessitates careful treatment of scalar multiplication, duality of vector spaces and maps, and tensor products of vector spaces and maps ([GM], [P]). The study of Hermitian forms seems to require a preliminary investigation into the fundamentals of the theory of bilinear forms, which now forms the first half of these notes. The payoff from the detailed treatment of bilinear forms will be the natural way in which the Hermitian case follows, in the second half.

Chapter 0 gives a brief review of elementary facts about vector spaces, as in a first college course; this should be prerequisite knowledge for most readers. Chapter 1 then sketches a review of notions of spaces of maps Hom(U, V), tensor products U ⊗ V, and direct sums U ⊕ V, and introduces some canonical linear maps, with the notation and basic concepts which will be used in all the subsequent Chapters. Chapter 2 starts with a definition of the usual trace of a map V → V, and then states definitions for the \underline{generalized trace} of maps V ⊗ U → V ⊗ W, or V → V ⊗ W, whose output is an element of Hom(U, W), or W, respectively. Many of the theorems can be viewed as linear algebra versions of more general statements in category theory, as considered by [JSV], [Maltsiniotis], [K], [PS], [Stolz-Teichner], [S].

Chapter 3 offers a similar basis-free approach to definitions, properties, and examples of a \underline{metric} on a vector space, and the trace, or contraction, with respect to a metric. The metrics are assumed to be non-degenerate, and finite-dimensionality is a consequence. The main construction is a generalization of the well-known inner product T r(A^{T} · B) on the space of matrices; the construction of Theorem 3.41 shows how a metric on Hom(U, V) is induced by arbitrary metrics on U and V, so that Hom(U, V) is isometric to U^{∗} ⊗ V with the induced tensor product metric. Chapter 4 develops the W -valued case of the trace with respect to a metric.

The basis-free approach is motivated in part by its usefulness in the geometry of vector bundles and structures on them, including bilinear and Hermitian forms, and almost complex structures. Important geometric applications include real vector bundles with Riemannian metrics, pseudo-Riemannian metrics (since definiteness is not assumed), or symplectic forms. The linear algebra results can be restated geometrically, with linear maps directly replaced by bundle morphisms, “distinguished non-zero element” by “nonvanishing section,” and in some cases, “𝕂” by “trivial line bundle.”

The plan is to proceed at an elementary pace, so that if the first few Lemmas in Chapter 1 make sense to the reader, then nothing more advanced will be encountered after that. In particular, the relationships with differential geometry and category theory can be ignored entirely by the uninterested reader and are mentioned here only in optional “Remarks.” It will be pointed out when the finite-dimensionality is used— for example, in the Theorems in Chapter 2 about the vector valued trace T r_{V;U,W}, V must be finite-dimensional, but U and W need not be.

CHAPTER 0 Review of Elementary Linear Algebra

0.1. Vector spaces

Definition 0.1. Given a set V, a field 𝕂, a binary operation + : V × V → V (addition), and a function · : 𝕂×V → V (scalar multiplication), V is a \underline{vector space} means that the operations have all of the following properties:

(1) Associative Law for Addition: For any u ∈ V and v ∈ V and w ∈ V, (u + v) + w = u + (v + w).

(2) Existence of a Zero Element: There exists an element 0_{V} ∈ V such that for any v ∈ V, v + 0_{V} = v.

(3) Existence of an Opposite: For each v ∈ V, there exists an element of V, called −v ∈ V, such that v + (−v) = 0_{V}.

(4) Associative Law for Scalar Multiplication: For any ρ, σ ∈ 𝕂 and v ∈ V, (ρσ) · v = ρ · (σ · v).

(5) Scalar Multiplication Identity: For any v ∈ V, 1 · v = v.

(6) Distributive Law: For all ρ, σ ∈ 𝕂 and v ∈ V, (ρ + σ) · v = (ρ · v) + (σ · v).

(7) Distributive Law: For all ρ ∈ 𝕂 and u, v ∈ V, ρ · (u + v) = (ρ · u) + (ρ · v). The following Exercises refer to a vector space V. Exercise 0.2 (Right Cancellation). Given u, v, w ∈ V, if u + w = v + w, then

u = v.

Hint. These first several Exercises, 0.2 through 0.11, can be proved using only the first three axioms about addition.

Exercise 0.3. Given u, w ∈ V, if u + w = w, then u = 0_{V}. Exercise 0.4. For any v ∈ V, (−v) + v = 0_{V}. Exercise 0.5. For any v ∈ V, 0_{V} + v = v. Exercise 0.6 (Left Cancellation). Given u, v, w ∈ V, if w + u = w + v, then

u = v.

Exercise 0.7 (Uniqueness of Zero Element). Given u, w ∈ V, if w + u = w, then u = 0_{V}.

Exercise 0.8 (Uniqueness of Additive Inverse). Given v, w ∈ V, if v + w = 0_{V} then v = −w and w = −v.

Exercise 0.9. −0_{V} = 0_{V}. Exercise 0.10. For any v ∈ V, −(−v) = v. Exercise 0.11. Given u, x ∈ V, −(u + x) = (−x) + (−u).

The previous results only used the properties of “+,” but the next result, even though its statement refers only to +, uses a scalar multiplication trick, together with the distributive axioms, which relate scalar multiplication to addition.

Theorem 0.12 (Commutative Property of Addition). For any v, w ∈ V,

v + w = w + v.

Proof. We start with this element of V, (1 + 1) · (v + w), and then set LHS=RHS, and use both distributive laws:

(1 + 1) · (v + w) = (1 + 1) · (v + w)
((1 + 1) · v) + ((1 + 1) · w) = (1 · (v + w)) + (1 · (v + w))
((1 · v) + (1 · v)) + ((1 · w) + (1 · w)) = (v + w) + (v + w)
(v + v) + (w + w) = (v + w) + (v + w).

Then, the associative law gives v + (v + (w + w)) = v + (w + (v + w)), and Left Cancellation leaves v + (w + w) = w + (v + w). Using the associative law again, (v+w)+w = (w+v)+w, and Right Cancellation gives the result v+w = w+v.

Exercise 0.13. For any v ∈ V, 0 · v = 0_{V}.

Exercise 0.14. For any v ∈ V, (−1) · v = −v.

Exercise 0.15. For any ρ ∈ 𝕂, ρ · 0_{V} = 0_{V}.

Exercise 0.16. For any ρ ∈ 𝕂 and u ∈ V, (−ρ) · u = −(ρ · u).

Exercise 0.17. Given ρ ∈ 𝕂 and u ∈ V, if ρ · u = 0_{V}, then ρ = 0 or u = 0_{V}.

Exercise 0.18. If \frac{1}{2} ∈ 𝕂, then for any v ∈ V, the following are equivalent. (1) v + v = 0_{V}, (2) v = −v, (3) v = 0_{V}.

Hint. Only the implication (1) ⟹ (3) requires \frac{1}{2} ∈ 𝕂; the others can be proved using only properties of +.

Definition 0.19. It is convenient to abbreviate the sum v + (−w) as v − w. This defines vector subtraction, so that \underline{v minus w} is defined to be the sum of v and the opposite (or “additive inverse”) of w.

Notation 0.20. Considering the associative law for addition, it is convenient to write the sum of more than two terms without all the parentheses: u + v + w can mean either (u + v) + w, or u + (v + w), since we get the same result either way. In light of Exercise 0.16, we can write −ρ · v to mean either (−ρ) · v or −(ρ · v), since these are the same. The multiplication “dot” can be used, or omitted, for both scalar times scalar and scalar times vector, when it is clear which symbols are scalars and which are vectors: instead of 3 · u, just write 3u. It is also convenient to establish an “order of operations,” so that scalar multiplication is done before addition or subtraction. So, 4v+u−3w is a short way to write (4·v)+(u+(−(3·w))).

0.2. Subspaces

The general idea of the statement “W is a subspace of V ” is that W is a vector space contained in a bigger vector space V with the same field 𝕂, and the + and · operations are the same in W as they are in V.

Definition 0.21. Let (V, +_{V} , ·_{V}) be a vector space with field of scalars 𝕂. A set W is a \underline{subspace} of V means:

• W ⊆ V, and

• There are operations +_{W} : W × W → W and ·_{W} : 𝕂 × W → W such that

(W, +_{W} , ·_{W}) is a vector space, and

• For all x, y ∈ W, x +_{V} y = x +_{W} y, and

• For all x ∈ W, ρ ∈ 𝕂, ρ ·_{V} x = ρ ·_{W} x.

Theorem 0.22. If W is a subspace of V, where V has zero element 0_{V}, then 0_{V} is an element of W, and is equal to the zero element of W.

Proof. By the second part of Definition 0.21, W is a vector space, so by Definition 0.1 applied to W, W contains a zero element 0_{W} ∈ W. By the first part of Definition 0.21, W ⊆ V, which implies 0_{W} ∈ V. By Definition 0.1 applied to W, 0_{W} +_{W} 0_{W} = 0_{W}, and by Definition 0.21, 0_{W} +_{V} 0_{W} = 0_{W} +_{W} 0_{W}. It follows that 0_{W} +_{V} 0_{W} = 0_{W} ∈ V, and then Exercise 0.3 implies 0_{W} = 0_{V}.

Theorem 0.22 can be used in this way: if W is a set that does \underline{not} contain 0_{V} as one of its elements, then W is \underline{not} a subspace of V.

Theorem 0.23. If W is a subspace of V, then for every w ∈ W, the opposite of w in W is the same as the opposite of w in V.

Proof. Let w be an element of W; then w ∈ V because W ⊆ V.

First, we show that an additive inverse of w in W is also an additive inverse of w in V. Let y be any additive inverse of w in W, meaning y ∈ W and w+_{W} y = 0_{W}. (There exists at least one such y, by Definition 0.1 applied to W.) W ⊆ V implies y ∈ V. From Theorem 0.22, 0_{W} = 0_{V}, and w +_{W} y = w +_{V} y by Definition 0.21, so w +_{V} y = 0_{V}, which means y is an additive inverse of w in V.

Second, we show that an additive inverse of w in V is also an additive inverse of w in W. Let z be any additive inverse of w in V, meaning z ∈ V and w +_{V} z = 0_{V}. (There exists at least one such z, by Definition 0.1 applied to V.) Then w +_{V} z = 0_{V} = w +_{V} y, so by Left Cancellation in V, z = y and y ∈ W, which imply z ∈ W and w +_{W} z = w +_{W} y = 0_{W}, meaning z is an additive inverse of w in W.

By uniqueness of opposites (Exercise 0.8 applied to either V or W), we can refer to y = z as “the” opposite of w, and denote it y = −w.

Theorem 0.23 also implies that subtraction in W is the same as subtraction in V: by Definition 0.19, for v, w ∈ W, v −_{W} w = v +_{W} y = v +_{V} y = v −_{V} w.

Theorem 0.23 can be used in this way: if W is a subset of a vector space V and there is an element w ∈ W, where the opposite of w in V is \underline{not} an element of W, then W is \underline{not} a subspace of V.

Theorem 0.24. Let (V, +_{V} , ·_{V}) be a vector space, and let W be a subset of V. Then W, with the same addition and scalar multiplication operations, is a subspace of V if and only if:

(1) x ∈ W, y ∈ W imply x +_{V} y ∈ W (closure under +_{V} addition), and

(2) ρ ∈ 𝕂, x ∈ W imply ρ ·_{V} x ∈ W (closure under ·_{V} scalar multiplication), and

(3) W ≠Ø.

Proof. Let V have zero element 0_{V}.

First suppose W is a subspace, so that as in the Proof of Theorem 0.22, W contains a zero element 0_{W}, which shows W ≠Ø, and (3) is true. From Property 1. of Definition 0.1, x ∈ W, y ∈ W imply x +_{W} y ∈ W, and from the definition of subspace, x +_{W} y = x +_{V} y, so x +_{V} y ∈ W, establishing (1). Similarly, from Property 5. of Definition 0.1, ρ ∈ 𝕂 implies ρ ·_{W} x ∈ W, and from the definition of subspace, ρ ·_{W} x = ρ ·_{V} x, so ρ ·_{V} x ∈ W, establishing (2).

Conversely, it follows from (1), (2), and (3) that W is a subspace of V, as follows: W is a subset of V by hypothesis. Define +_{W} and ·_{W} by x +_{W} y = x +_{V} y, and ρ ·_{W} x = ρ ·_{V} x — these define operations on W by (1) and (2) (the closure Properties 1. and 5. from Definition 0.1, and also parts of Definition 0.21), but it remains to check the other properties to show that (W, +_{W} , ·_{W}) is a vector space. Since W ≠Ø by (3), there is some x ∈ W, and by (2), 0·_{V} x ∈ W. By Exercise 0.13, 0·_{V} x = 0_{V}, so 0_{V} ∈ W, and it satisfies x+_{W} 0_{V} = x+_{V} 0_{V} = x for all x ∈ W, so 0_{V} is a zero element for W. The scalar multiple identity also works: 1·_{W} x = 1·_{V} x = x. Also by (2), for any x ∈ W, (−1) ·_{V} x ∈ W, and it is easy to check (−1) ·_{V} x is an additive inverse of x in W: x +_{W} ((−1) ·_{V} x) = (1 ·_{V} x) +_{V} ((−1) ·_{V} x) = (1 + (−1)) ·_{V} x = 0 ·_{V} x = 0_{V}. The other vector space properties, (2,6,8,9) from Definition 0.1, follow immediately from the facts that these properties hold in V and the operations in W give the same sums and scalar multiples.

Definition 0.25. Given a vector space V and any subset S ⊆ V, the \underline{span} of S is the subset of V defined as the set of all (finite) sums of elements of S with coefficients in 𝕂:

{α_{1}s_{1} + α_{2}s_{2} + … + α_{ν}s_{ν} : α_{1}, …, α_{ν} ∈ 𝕂, s_{1}, …, s_{ν} ∈ S}.

This is always a subspace of V. (We define the span of Ø to be {0_{V} }.)

Definition 0.26. A vector space V is \underline{finite-dimensional} means that there exists a finite subset S ⊆ V such that the span of S is equal to V.

Definition 0.27. Given an ordered list (possibly with repeats) of elements of a vector space V, (u_{1}, u_{2}, …, u_{ν}), the list is \underline{linearly independent} means that the following implication holds for scalars α_{1}, …, α_{ν} ∈ 𝕂:

α_{1}u_{1} + α_{2}u_{2} + … + α_{ν}u_{ν} = 0_{V} ⟹ α_{1} = α_{2} = … = α_{ν} = 0.

The empty list is linearly independent. We will not have occasion to consider infinite lists.

Claim 0.28. If V is a vector space equal to the span of a finite set S with ν elements, then there is no linearly independent list of elements of V of length > ν.

Definition 0.29. Given an ordered list of elements of a vector space V, (u_{1}, u_{2}, …, u_{ν}), the list is an \underline{ordered basis} of V means that the list is linearly independent and the span of the set {u_{1}, u_{2}, …, u_{ν}} is equal to V.

0.3. Additive functions and linear functions

Let U, V, and W be vector spaces with the same field of scalars 𝕂.

Definition 0.30. A function F : U → V is \underline{additive} means: F has the property that F (u + w) = F (u) + F (w) for all u, w ∈ U.

Definition 0.31. A function F : U → V is \underline{linear} means: F is additive and also has the \underline{scaling property}: F (ρ · u) = ρ · F (u) for all ρ ∈ 𝕂 and all u ∈ U.

Exercise 0.32. If F : U → V is additive, then: F (0_{U}) = 0_{V}, and for all u ∈ U, F (−u) = −F (u).

Exercise 0.33. If 𝕂 contains the rational numbers ℚ as a subrng, and F : U → V is additive, then for every rational number ρ ∈ ℚ, F (ρ · u) = ρ · F (u).

Hint. Start by showing that for integers ν ∈ ℤ, F (ν · u) = ν · F (u).

Exercise 0.34. Give an example of a field, vector spaces V and W, and a function F : V → W, such that F has the scaling property, but which is not linear because it is not additive.

Exercise 0.35. Give an example of a field, vector spaces V and W, and a function F : V → W such that F has the additive property, but which is not linear because it does not have the scaling property.

Hint. Try this for 𝕂 = ℂ and then for 𝕂 = ℝ (harder).

Notation 0.36. At this point, the arrow notation F : V → W will only be used for functions that are linear from a vector space V (the \underline{domain}) to another vector space W (the \underline{target}). A linear function will also be called a \underline{map}, linear map, or \underline{arrow}, or when it is convenient to emphasize the scalar field, a \underline{𝕂-linear} map. A function which is not necessarily linear will be denoted with a variant ⇝ arrow symbol.

Example 0.37. Given a vector space V and a subspace W, the canonical \underline{subspace inclusion} function Q : W → V defined by Q(w) = w is linear.

Notation 0.38. As a special case of the above inclusion map (and as in Example 6.15), the \underline{identity map} Id_{V} : V → V, defined by the formula Id_{V} (v) = v, is linear.

Exercise 0.39. Given vector spaces U, V, W, and functions F : W → V, G : V → U, if F and G are linear, then the composite G ∘ F : W → U is linear.

The ∘ notation for composites is as in Notation 6.16.

Example 0.40. For a linear function F : W → V, the \underline{image} of F is the set F (W) = {F (w) ∈ V : w ∈ W }. The image is always a subspace of the target V.

Example 0.41. For a linear function F : W → V, the \underline{kernel} of F is the set ker(F) = {w ∈ W : F (w) = 0_{V} }. The kernel is always a subspace of the domain W.

Definition 0.42. A linear map C : X → Y is a \underline{linear monomorphism} means: C has the following cancellation property for any compositions with linear maps A and B (which are well-defined in the sense that X is the target space of both A and B),

C ∘ A = C ∘ B ⟹ A = B.

Exercise 0.43. Given a linear map F : U → V, the following are equivalent.

(1) F is one-to-one.

(2) F is left cancellable.

(3) F has a left inverse: there exists a function H : V ⇝ U such that H ∘ F = Id_{U}.

(4) F is a linear monomorphism.

(5) ker(F) = {0_{U}}.

Hint. The first three properties are considered in Exercise 6.17, and their equivalence is a matter of set theory only, without using the linearity of F. In particular, the function H is not necessarily linear. The implication (2) ⟹ (4) is trivial, and (1) ⟹ (5) uses only F (0_{U}) = 0_{V} from Exercise 0.32. The implications (5) ⟹ (1) and (4) ⟹ (5) follow from the linearity of F.

Exercise 0.44. If V is finite-dimensional and F : U → V is linear and one-to-one, then U is finite-dimensional.

Hint. This follows from Claim 0.28 and the fact that an independent list in U is transformed to an independent list in V by a one-to-one map.

Exercise 0.45. Given linear maps A : W → V and F : U → V, if A(W) ⊆ F (U) and H : V ⇝ U is any left inverse of F, then H ∘ A : W → U is linear.

Hint. For any two elements v_{1}, v_{2} ∈ F (U), v_{1} = F (u_{1}) for some unique u_{1} as in Exercise 0.43, and similarly v_{2} = F (u_{2}). Using the linearity of F and any scalar λ,

H(v_{1} + v_{2}) = H(F (u_{1}) + F (u_{2})) = H(F (u_{1} + u_{2}))
(0.1)  = u_{1} + u_{2} = H(F (u_{1})) + H(F (u_{2})) = H(v_{1}) + H(v_{2}),
H(λ · v_{1}) = H(λ · F (u_{1})) = H(F (λ · u_{1}))
= λ · u_{1} = λ · H(F (u_{1})) = λ · H(v_{1}).

In particular, for w_{1}, w_{2} ∈ W, let A(w_{1}) = F (u_{1}) and A(w_{2}) = F (u_{2}). Then, using the additivity of A and (0.1),

(H ∘ A)(w_{1} + w_{2}) = H(A(w_{1}) + A(w_{2})) = (H ∘ A)(w_{1}) + (H ∘ A)(w_{2}).

The scaling property for H ∘ A follows similarly.

Definition 0.46. A map C : X → Y is a \underline{linear epimorphism} means: C has the following cancellation property for any compositions with linear maps A and B,

A ∘ C = B ∘ C ⟹ A = B.

Exercise 0.47. If the linear map F : U → V has a right inverse, meaning that there exists H : V ⇝ U so that F ∘ H = Id_{V}, then F is onto and right cancellable as in Exercise 6.18, and the right cancellable property implies that F is a linear epimorphism.

Exercise 0.48. Given vector spaces V, W, and a linear function F : W → V, the following are equivalent.

(1) F is both one-to-one and onto.

(2) F has a right inverse H_{1} : V ⇝ W and a left inverse H_{2} : V ⇝ W.

(3) There exists a linear map G : V → W such that G is a left inverse of F and G is a right inverse of F.

Hint. The (3) ⟹ (2) direction is trivial. The equivalence (1) ⟺ (2) does not use the linearity of F — see Exercise 6.20, which also shows that (2) implies H_{1} = H_{2}, so G : V ⇝ W in (3) can be chosen to equal H_{1} = H_{2}. The linearity of F then implies the linearity of G by Exercise 0.45.

Notation 0.49. Given vector spaces V, W, and a linear function F : W → V, F is \underline{invertible} means that F satisfies any of the three equivalent properties (1), (2), or (3) from Exercise 0.48, as in Definition 6.21. However, we usually use property (3), so that as in Notation 6.22, G is the unique \underline{inverse} of F, denoted G = F^{−1}. It follows that F^{−1} is also invertible, with inverse F.

Claim 0.50. If V is finite-dimensional and F : V → V is linear and F is either a linear monomorphism or a linear epimorphism, then F is invertible.

Claim 0.51. Given a finite-dimensional vector space V and a linear map F : V → V, the following are equivalent.

(1) For all linear maps A : V → V, A ∘ F = F ∘ A : V → V.

(2) There exists a scalar λ ∈ 𝕂 so that for all v ∈ V, F (v) = λ · v.

Proof. The second property can be denoted F = λ · Id_{V}. See [B] Exercise II.1.26 or [J] §3.11.

CHAPTER 1 Abstract Linear Algebra

1.1. Spaces of linear maps

For this Chapter, we fix an arbitrary field (𝕂, +, ·, 0, 1). All vector spaces use the same scalar field 𝕂, and + and · also refer to the vector space addition and scalar multiplication. A 𝕂-linear map A with domain U and target V, such that A(u) = v, will be written as A : U → V : u ↦ v, or the A may appear near the arrow when several maps are combined in a diagram.

Notation 1.1. The set of all 𝕂-linear maps from U to V is denoted Hom(U, V).

Claim 1.2. Hom(U, V) is itself a vector space over 𝕂. If U and V are finitedimensional, then Hom(U, V) is finite-dimensional.

Notation 1.3. Hom(V, V) is abbreviated End(V), the space of endomorphisms of V. The space End(V) has a distinguished element, the identity map denoted Id_{V} : v ↦ v from Notation 0.38.

Notation 1.4. Hom(V, 𝕂) is abbreviated V^{∗}, the \underline{dual space} of V.

Definition 1.5. For maps A : U′ → U and B : V → V′, define

Hom(A, B) : Hom(U, V) → Hom(U′, V′)

so that for F : U → V,

Hom(A, B)(F) = B ∘ F ∘ A : U′ → V′.

Lemma 1.6. ([B] §II.1.2) If A : U → V, B : V → W, C : X → Y, D : Y → Z, then

Hom(A, D) ∘ Hom(B, C) = Hom(B ∘ A, D ∘ C) : Hom(W, X) → Hom(U, Z).

Definition 1.7. For any vector spaces U, V, W, define a \underline{generalized transpose} map,

t_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W), Hom(U, W)),

so that for A : U → V, B : V → W,

t_{UV}^{W} (A) = Hom(A, Id_{W}) : B ↦ B ∘ A.

Lemma 1.8. For any vector spaces U, V, W, U′, V′, W′, and any maps E : U′ → U, F : V → V′, G : W → W′, the following diagram is commutative.

\xymatrix{Hom(U, V) \ar[r]^{t_{UV}^{W}} \ar[dd]_{Hom(E,F)} & Hom(Hom(V, W), Hom(U, W)) \ar[d]^{Hom(Hom(F,Id_{W}),Hom(E,G))} \\  & Hom(Hom(V′, W), Hom(U′, W′)) \\ Hom(U′, V′) \ar[r]^{t_{U′V ′}^{W′}} & Hom(Hom(V′, W′), Hom(U′, W′)) \ar[u]_{Hom(Hom(Id_{V ′} ,G),Hom(Id_{U′} ,Id_{W′}))}}

Proof. For any A : U → V,

Hom(Hom(Id_{V′}, G), Hom(Id_{U′}, Id_{W′})) ∘ t_{U′V′}^{W′} ∘ Hom(E, F) :
A ↦ (t_{U′V′}^{W′}(F ∘ A ∘ E)) ∘ Hom(Id_{V′}, G)
= Hom(F ∘ A ∘ E, Id_{W′}) ∘ Hom(Id_{V′}, G),
Hom(Hom(F, Id_{W}), Hom(E, G)) ∘ t_{UV}^{W} :
A ↦ Hom(E, G) ∘ Hom(A, Id_{W}) ∘ Hom(F, Id_{W}).

The claimed equality follows from Lemma 1.6.

Notation 1.9. In the special case W = 𝕂, t_{UV}^{𝕂} is a canonical transpose map from Hom(U, V) to Hom(V^{∗}, U^{∗}), and it is abbreviated t_{UV}^{𝕂} = t_{UV}.

Notation 1.10. t_{UV} (A) = Hom(A, Id_{𝕂}) is abbreviated by A^{∗} : V^{∗} → U^{∗}, so that for ϕ ∈ V^{∗}, A^{∗}(ϕ) is the map ϕ ∘ A : U → 𝕂, i.e., (A^{∗}(ϕ))(u) = ϕ(A(u)).

Claim 1.11. For any vector spaces U, V, the map

t_{UV} : Hom(U, V) → Hom(V^{∗}, U^{∗}) : A ↦ A^{∗}

is one-to-one. If V is finite-dimensional then t_{UV} is invertible.

Lemma 1.12. For A : U → V and F : V → V′, (F ∘ A)^{∗} = A^{∗} ∘ F^{∗}, and if A is invertible, then so is A^{∗}, with (A^{−1})^{∗} = (A^{∗})^{−1}. Also, Id_{V^{∗}} = Id_{V}^{∗}.

Proof. The claim about F ∘ A follows from Lemma 1.8 (with E = Id_{U}, G = Id_{𝕂}), or by applying Lemma 1.6 directly. Note that t_{V V} : End(V) → End(V^{∗}) takes the distinguished element Id_{V} ∈ End(V) to the distinguished element t_{V V} (Id_{V}) = Id_{V}^{∗} = Id_{V^{∗}} ∈ End(V^{∗}).

Definition 1.13. For any vector spaces V, W, define

d_{V W} : V → Hom(Hom(V, W), W)

so that for v ∈ V, H ∈ Hom(V, W),

(d_{V W} (v)) : H ↦ H(v).

Lemma 1.14. For any vector spaces U, V, W, X and any maps H : U → V, G : W → X, the following diagram is commutative.

\xymatrix{U \ar[rr]^{d_{UW}} \ar[dd]_{H} \ar[dr]^{d_{UX}} &  & Hom(Hom(U, W), W) \ar[d]^{Hom(Hom(H,Id_{W}),G)} \\  & Hom(Hom(U, X), X) \ar[r]_{Hom(Hom(H,G),Id_{X})} & Hom(Hom(V, W), X) \\ V \ar[rr]^{d_{V X}} &  & Hom(Hom(V, X), X) \ar[u]_{Hom(Hom(Id_{V} ,G),Id_{X})}}

Proof. For u ∈ U, A : V → W,

Hom(Hom(Id_{V} , G), Id_{X}) ∘ d_{V X} ∘ H : u ↦ (d_{V X}(H(u))) ∘ Hom(Id_{V} , G) :
A ↦ (G ∘ A)(H(u)),
Hom(Hom(H, G), Id_{X}) ∘ d_{UX} : u ↦ (d_{UX}(u)) ∘ Hom(H, G) :
A ↦ (G ∘ A ∘ H)(u),
Hom(Hom(H, Id_{W}), G) ∘ d_{UW} : u ↦ G ∘ (d_{UW} (u)) ∘ Hom(H, Id_{W}) :
A ↦ G((d_{UW} (u))(A ∘ H))
= G((A ∘ H)(u)).

As a special case of Lemma 1.14 with W = X, G = Id_{W}, for any H : U → V,

(1.1)  (t_{Hom(V,W),Hom(U,W)}^{W}(t_{UV}^{W} (H))) ∘ d_{UW} = d_{V W} ∘ H,

where t_{Hom(V,W),Hom(U,W)}^{W}(t_{UV}^{W} (H)) = Hom(Hom(H, Id_{W}), Id_{W}) by Definition 1.7 of the t maps.

Notation 1.15. In the special case W = 𝕂, d_{V 𝕂} is abbreviated d_{V}. It is the canonical \underline{double duality} map d_{V} : V → V^{∗∗}, defined by (d_{V} (v))(ϕ) = ϕ(v).

The case (1.1) of Lemma 1.14 then gives the equation ([B] §II.2.7, [AF] §20)

(1.2)  d_{V} ∘ A = A^{∗∗} ∘ d_{U},

where for A : U → V, A^{∗∗} abbreviates t_{V^{∗}U^{∗}}(t_{UV} (A)).

Claim 1.16. The canonical map d_{V} is one-to-one. d_{V} is invertible if and only if V is finite-dimensional.

Proof. The one-to-one property is easily checked. See [B] §II.7.5.

Lemma 1.17. Hom(d_{V W} , Id_{W}) ∘ d_{Hom(V,W),W} = Id_{Hom(V,W)}.

Proof. For v ∈ V, K : V → W,

((Hom(d_{V W} , Id_{W}) ∘ d_{Hom(V,W),W})(K))(v) = (d_{Hom(V,W),W} (K))(d_{V W} (v))
= (d_{V W} (v))(K) = K(v).

In the W = 𝕂 case, this one-sided inverse relation gives ([AF] §20)

d_{V}^{∗} ∘ d_{V^{∗}} = Id_{V^{∗}}.

Remark 1.18. In some applications, the vector space V is often identified with a subspace of V^{∗∗}, or the map d_{V} is ignored, but it is less trouble than might be expected to keep V and V^{∗∗} distinct, and always accounting for d_{V} turns out to be convenient bookkeeping.

Lemma 1.19. Suppose U is finite-dimensional. For A : U → V, if A^{∗} has a linear right inverse F : U^{∗} → V^{∗}, so that A^{∗} ∘ F = Id_{U^{∗}}, then A has a linear left inverse. If A^{∗} has a linear left inverse E : U^{∗} → V^{∗}, so that E ∘ A^{∗} = Id_{V^{∗}}, then A has a linear right inverse.

Proof. Using Lemma 1.12, case (1.2) of Lemma 1.14, and Claim 1.16,

d_{U}^{−1} ∘ F^{∗} ∘ d_{V} ∘ A = d_{U}^{−1} ∘ F^{∗} ∘ A^{∗∗} ∘ d_{U} = d_{U}^{−1} ∘ (A^{∗} ∘ F)^{∗} ∘ d_{U} = Id_{U}.

Using the one-to-one property of d_{V},

d_{V} ∘ A ∘ d_{U}^{−1} ∘ E^{∗} ∘ d_{V} = A^{∗∗} ∘ E^{∗} ∘ d_{V} = (E ∘ A^{∗})^{∗} ∘ d_{V} = d_{V}
⟹ A ∘ d_{U}^{−1} ∘ E^{∗} ∘ d_{V} = Id_{V} .

Definition 1.20. For any vector space W, define m : W → Hom(𝕂, W) so that for w ∈ W,

m(w) : λ ↦ λ · w.

Lemma 1.21. For any vector spaces W, W′, with two maps m, m′ as indicated, and any maps ϕ ∈ 𝕂^{∗}, G : W → W′, the following diagram is commutative.

\xymatrix{W \ar[r]^{m} \ar[d]_{ϕ(1)·G} & Hom(𝕂, W) \ar[d]^{Hom(ϕ,G)} \\ W′ \ar[r]^{m′} & Hom(𝕂, W′).}

Proof. For λ ∈ 𝕂, w ∈ W,

Hom(ϕ, G) ∘ m : w ↦ G ∘ (m(w)) ∘ ϕ : λ ↦ G(ϕ(λ) · w),
m′ ∘ (ϕ(1) · G) : w ↦ m′(ϕ(1) · G(w)) : λ ↦ λ · ϕ(1) · G(w).

Lemma 1.22. For any vector space W, m : W → Hom(𝕂, W) is invertible.

Proof. An inverse is

m^{−1} = d_{𝕂W} (1) : Hom(𝕂, W) → W : A ↦ A(1).

Checking both composites, ((m ∘ m^{−1})(A))(λ) = λ · A(1) = A(λ), and (m^{−1} ∘ m)(w) = (m(w))(1) = 1 · w = w.

1.2. Tensor products

Definition 1.23. For vector spaces U, V, W, a function A : U × V ⇝ W is a \underline{bilinear function} means: for any λ ∈ 𝕂, u_{1}, u_{2} ∈ U, v_{1}, v_{2} ∈ V,

A(u_{1} + u_{2}, v_{1}) = A(u_{1}, v_{1}) + A(u_{2}, v_{1}),
A(u_{1}, v_{1} + v_{2}) = A(u_{1}, v_{1}) + A(u_{1}, v_{2}), and
A(λ · u_{1}, v_{1}) = A(u_{1}, λ · v_{1}) = λ · A(u_{1}, v_{1}).

Remark 1.24. We remark that the above Definition is different from the notion of \underline{bilinear form}, from Definition 3.1 in Chapter 3. The way in which these types of functions are related is described in Example 1.51.

For any vector spaces U and V, there exists a \underline{tensor product} vector space U ⊗ V, which we informally define as the set of formal finite sums

(1.3)  ρ_{1} · u_{1} ⊗ v_{1} + ρ_{2} · u_{2} ⊗ v_{2} + ⋯ + ρ_{ν} · u_{ν} ⊗ v_{ν},

for scalars ρ_{1}, …, ρ_{ν} ∈ 𝕂, elements u_{1}, …, u_{ν} ∈ U, and elements v_{1}, …, v_{ν} ∈ V, with addition and scalar multiplication carried out in the usual way, subject to the relations:

(u_{1} + u_{2}) ⊗ v = u_{1} ⊗ v + u_{2} ⊗ v,
u ⊗ (v_{1} + v_{2}) = u ⊗ v_{1} + u ⊗ v_{2},
(ρ · u) ⊗ v = u ⊗ (ρ · v) = ρ · (u ⊗ v).

A term u ⊗ v is also called a \underline{tensor product} of the vectors u and v. The operation taking a pair of vectors to their tensor product is a bilinear function, denoted

τ : U × V ⇝ U ⊗ V : (u, v) ↦ u ⊗ v.

U ⊗ V and τ have the property that for any bilinear function A : U × V ⇝ W as in Definition 1.23, there is a unique linear map a : U ⊗ V → W such that A = a ∘ τ, that is, for any u ∈ U and v ∈ V,

(1.4)  A(u, v) = a(u ⊗ v).

So bilinear functions can be converted to linear maps, by replacing the domain U × V with the tensor product U ⊗ V.

A more formal proof of the existence of U ⊗ V and τ as above uses methods different from the main stream of this Chapter, so we refer to Appendix 6.3, where (1.4) is stated as Theorem 6.35. The following re-statement of the description (1.3) also follows from the construction of Appendix 6.3:

Lemma 1.25. For any U, V, there exists a space U ⊗ V and a bilinear function τ : U × V ⇝ U ⊗ V such that U ⊗ V is equal to the span of elements of the form τ (u, v) = u ⊗ v. Any linear map B : U ⊗ V → W is uniquely determined by its values on the set of elements u ⊗ v.

Remark 1.26. In general, although the set of tensor products of vectors, {u⊗v : u ∈ U, v ∈ V } spans the space U ⊗ V, not every element of U ⊗ V is of the form u ⊗ v — generally elements are finite sums as in (1.3).

Lemma 1.27. If U and V are finite-dimensional, then so is U ⊗ V.

Example 1.28. The scalar multiplication operation ·_{U} : 𝕂×U ⇝ U is a bilinear function, and induces a map

l_{U} : 𝕂 ⊗ U → U : λ ⊗ u ↦ λ · u.

The map l_{U} is invertible, with inverse l_{U}^{−1}(u) = 1 ⊗ u, and sometimes l_{U} is abbreviated l, with the same l notation used for maps U ⊗ 𝕂 → U : u ⊗ λ ↦ λ · u.

Example 1.29. The switching function

S : U × V ⇝ V × U : (u, v) ↦ (v, u)

composes with τ : V × U ⇝ V ⊗ U:

τ ∘ S : U × V ⇝ V ⊗ U : (u, v) ↦ v ⊗ u.

This composite is a bilinear function and induces an invertible, 𝕂-linear \underline{switching} \underline{map}

s : U ⊗ V → V ⊗ U : u ⊗ v ↦ v ⊗ u.

Claim 1.30. The spaces (U ⊗ V) ⊗ W and U ⊗ (V ⊗ W) are related by a canonical, invertible, 𝕂-linear map.

Remark 1.31. In addition to the discussion in Appendix 6.3, the above statements about tensor products follow standard constructions as in references [MB] §IX.8, [AF] §19, [K] §II.1. From this point, product sets U × V and the function τ will not appear often, but could be used by the reader to verify certain maps are well-defined. Lemma 1.25 will be used frequently and without comment when defining maps (as already done in Examples 1.28 and 1.29) or proving equality of two maps (as in Lemma 1.36, below).

Remark 1.32. The notion of a multilinear map is also standard, and the spaces from Claim 1.30 can be identified with each other and yet another space, a triple tensor product U⊗V ⊗W. This is spanned by elements of the form u⊗v⊗w, and it is a convenient abbreviation for our purposes to also identify elements (u ⊗ v) ⊗ w = u ⊗ (v ⊗ w) = u ⊗ v ⊗ w. We leave the justification for this to the references; some applications or generalizations of linear algebra keep track of this associativity and do not use such abbrevations, but we will make these identifications without comment. Similarly, notions of associativity for tensor products of more than three spaces or vectors will be implicitly assumed as needed.

A tensor product of 𝕂-linear maps canonically induces a 𝕂-linear map between tensor product spaces, as follows.

Definition 1.33. For any vector spaces U_{1}, U_{2}, V_{1}, V_{2}, define
j : Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) → Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}),

so that for A : U_{1} → V_{1}, B : U_{2} → V_{2}, u ∈ U_{1}, and v ∈ U_{2}, the map j(A ⊗ B) acts as:

(j(A ⊗ B)) : u ⊗ v ↦ (A(u)) ⊗ (B(v)).

Claim 1.34. The canonical map j is one-to-one. If one of the ordered pairs (U_{1}, U_{2}), (U_{1}, V_{1}), or (U_{2}, V_{2}) consists of finite-dimensional spaces, then j is invertible.

Proof. See [B] §II.7.7 or [K] §II.2.

Although it frequently occurs in the literature that j(A ⊗ B) and A ⊗ B are identified, we will maintain the distinction. In this Section and later (many times in Section 2.2), the canonical map j will appear in diagrams as a function in its own right, and it is useful to keep track of it everywhere, instead of only ignoring it sometimes. However, the notation j(A ⊗ B) is not always as convenient as the following abbreviation.

Notation 1.35. Let [A⊗B] denote j(A⊗B), so that, for example, the equation

(j(A ⊗ B))(u ⊗ v) = (A(u)) ⊗ (B(v))

appears as

[A ⊗ B](u ⊗ v) = (A(u)) ⊗ (B(v)).

Lemma 1.36. ([B] §II.3.2, [K] §II.6)

[A ⊗ B] ∘ [E ⊗ F] = [(A ∘ E) ⊗ (B ∘ F)].

Note that the brackets conveniently establish an order of operations, and appear three times in the Lemma, but may stand for three distinct canonical j maps, depending on the domains of A, B, E, and F. When it is necessary or convenient to keep track of different maps, the j symbols are used instead of the brackets, and are sometimes labeled with subscripts, primes, etc., as in the following Lemma.

Lemma 1.37. For any vector spaces U_{1}, U_{2}, U_{3}, U_{4}, V_{1}, V_{2}, V_{3}, V_{4}, with maps j′, j′′ as indicated, and any maps A_{1} : U_{3} → U_{1}, A_{2} : U_{4} → U_{2}, B_{1} : V_{1} → V_{3}, B_{2} : V_{2} → V_{4}, the following diagram is commutative.

\xymatrix{Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) \ar[d]^{[Hom(A_{1},B_{1})⊗Hom(A_{2},B_{2})]} \ar[r]^{j′} & Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) \ar[d]^{Hom([A_{1}⊗A_{2}],[B_{1}⊗B_{2}])} \\ Hom(U_{3}, V_{3}) ⊗ Hom(U_{4}, V_{4}) \ar[r]^{j′′} & Hom(U_{3} ⊗ U_{4}, V_{3} ⊗ V_{4})}

Proof. For E : U_{1} → V_{1} and F : U_{2} → V_{2},
E ⊗ F ↦ (j′′ ∘ [Hom(A_{1}, B_{1}) ⊗ Hom(A_{2}, B_{2})])(E ⊗ F)
= j′′((Hom(A_{1}, B_{1})(E)) ⊗ (Hom(A_{2}, B_{2})(F)))
= j′′((B_{1} ∘ E ∘ A_{1}) ⊗ (B_{2} ∘ F ∘ A_{2})),
E ⊗ F ↦ (Hom([A_{1} ⊗ A_{2}], [B_{1} ⊗ B_{2}]) ∘ j′)(E ⊗ F)
= [B_{1} ⊗ B_{2}] ∘ (j′(E ⊗ F)) ∘ [A_{1} ⊗ A_{2}]
= j′′((B_{1} ∘ E ∘ A_{1}) ⊗ (B_{2} ∘ F ∘ A_{2})).

The last step uses Lemma 1.36.

Lemma 1.38. For any vector spaces U, W, and any maps ϕ ∈ 𝕂^{∗}, F ∈ Hom(U, W), the following diagram is commutative.

\xymatrix{𝕂 ⊗ U \ar[d]_{[ϕ⊗F]} \ar[r]^{l_{U}} & U \ar[d]^{ϕ(1)·F} \\ 𝕂 ⊗ W \ar[r]^{l_{W}} & W}

Proof. For λ ∈ 𝕂, u ∈ U,

l_{W} ∘ [ϕ ⊗ F] : λ ⊗ u ↦ l_{W} ((ϕ(λ)) ⊗ (F (u))) = ϕ(λ) · F (u),
(ϕ(1) · F) ∘ l_{U} : λ ⊗ u ↦ ϕ(1) · F (λ · u).

Definition 1.39. For arbitrary vector spaces U, V, W, define

n : Hom(U, V) ⊗ W → Hom(U, V ⊗ W)

so that for A : U → V, w ∈ W, u ∈ U,

n(A ⊗ w) : u ↦ (A(u)) ⊗ w.

Notation 1.40. The ordering of the spaces from Definition 1.39 is not canonical; the “n” label (with various subscripts) is used for analogously defined maps:

n_{0} : Hom(U, V) ⊗ W → Hom(U, V ⊗ W) : A ⊗ w : (u ↦ (A(u)) ⊗ w)
n_{1} : Hom(U, V) ⊗ W → Hom(U, W ⊗ V) : A ⊗ w : (u ↦ w ⊗ (A(u)))
n_{2} : W ⊗ Hom(U, V) → Hom(U, V ⊗ W) : w ⊗ A : (u ↦ (A(u)) ⊗ w)
n_{3} : W ⊗ Hom(U, V) → Hom(U, W ⊗ V) : w ⊗ A : (u ↦ w ⊗ (A(u))).

For map with an n label appearing in a diagram or equation, its type from the above list of four formulas can usually be determined by context.

Lemma 1.41. For any vector spaces U, U′, V, V′, W, W′, with maps n, n′ as indicated, and any maps F : U′ → U, B : V → V′, C : W → W′, the following diagram is commutative.

\xymatrix{Hom(U, V) ⊗ W \ar[r]^{n} \ar[d]_{[Hom(F,B)⊗C]} & Hom(U, V ⊗ W) \ar[d]^{Hom(F,[B⊗C])} \\ Hom(U′, V′) ⊗ W′ \ar[r]^{n′} & Hom(U′, V′ ⊗ W′)}

Proof. For A : U → V, w ∈ W, u′ ∈ U′,

A ⊗ w ↦ (Hom(F, [B ⊗ C]) ∘ n)(A ⊗ w) = [B ⊗ C] ∘ (n(A ⊗ w)) ∘ F :
u′ ↦ [B ⊗ C]((A(F (u′))) ⊗ w) = ((B ∘ A ∘ F)(u′)) ⊗ (C(w)),
A ⊗ w ↦ (n′ ∘ [Hom(F, B) ⊗ C])(A ⊗ w) = n′((B ∘ A ∘ F) ⊗ (C(w))) :
u′ ↦ ((B ∘ A ∘ F)(u′)) ⊗ (C(w)).

The n map is related to a canonical j map.

Lemma 1.42. The following diagram is commutative.

\xymatrix{Hom(U, V) ⊗ W \ar[r]^{n} \ar[d]_{[Id_{Hom(U,V )}⊗m]} & Hom(U, V ⊗ W) \ar[d]^{Hom(l_{U} ,Id_{V ⊗W})} \\ Hom(U, V) ⊗ Hom(𝕂, W) \ar[r]^{j} & Hom(U ⊗ 𝕂, V ⊗ W)}

Proof. Starting with A ⊗ w ∈ Hom(U, V) ⊗ W,

A ⊗ w ↦ (j ∘ [Id_{Hom(U,V)} ⊗ m])(A ⊗ w) = j(A ⊗ (m(w))) :
u ⊗ λ ↦ (A(u)) ⊗ (λ · w) = λ · (A(u)) ⊗ w,
A ⊗ w ↦ (Hom(l_{U}, Id_{V ⊗W}) ∘ n)(A ⊗ w) = (n(A ⊗ w)) ∘ l_{U} :
u ⊗ λ ↦ (A(λ · u)) ⊗ w = λ · (A(u)) ⊗ w.

Lemma 1.43. The canonical map n is one-to-one, and if U or W is finitedimensional, then n is invertible.

Proof. This follows from Lemma 1.22, Claim 1.34, Lemma 1.42, and the invertibility of the l_{U} map as in Example 1.28. See also [B] §II.7.7 or [AF] §20.

Lemma 1.41, Lemma 1.42, and Lemma 1.43 all generalize in a straightforward way to the variant n maps as in Notation 1.40.

Definition 1.44. For arbitrary vector spaces U, V, W define

q : Hom(V, Hom(U, W)) → Hom(V ⊗ U, W)

so that for K : V → Hom(U, W), v ∈ V, u ∈ U,

(q(K))(v ⊗ u) = (K(v))(u).

Lemma 1.45. For any vector spaces U, V, W, q is invertible.

Proof. For D ∈ Hom(V ⊗ U, W), check that

q^{−1}(D) : v ↦ (u ↦ D(v ⊗ u))

defines an inverse. See also [AF] §19, [B] §II.4.1, [MB] §IX.11, or [K] §II.1.

Remark 1.46. In some applications, the map q^{−1} is called a “currying” transformation, and so q is an “uncurrying” map.

Lemma 1.47. ([AF] §20) For any vector spaces U_{1}, V_{1}, W_{1}, U_{2}, V_{2}, W_{2}, with maps q_{1}, q_{2} as indicated, and any maps D : V_{2} → V_{1}, E : U_{2} → U_{1}, F : W_{1} → W_{2}, the following diagram is commutative.

\xymatrix{Hom(V_{1}, Hom(U_{1}, W_{1})) \ar[r]^{q_{1}} \ar[d]_{Hom(D,Hom(E,F))} & Hom(U_{1} ⊗ V_{1}, W_{1}) \ar[d]^{Hom([E⊗D],F)} \\ Hom(V_{2}, Hom(U_{2}, W_{2})) \ar[r]^{q_{2}} & Hom(U_{2} ⊗ V_{2}, W_{2})}

Proof. Starting with any G : V_{1} → Hom(U_{1}, W_{1}),

(Hom([E ⊗ D], F) ∘ q_{1})(G) : u ⊗ v ↦ (F ∘ (q_{1}(G)) ∘ [E ⊗ D])(u ⊗ v)
= F ((G(D(v)))(E(u))),
(q_{2} ∘ Hom(D, Hom(E, F)))(G) : u ⊗ v ↦ (q_{2}(Hom(E, F) ∘ G ∘ D))(u ⊗ v)
= ((Hom(E, F) ∘ G ∘ D)(v))(u)
= (F ∘ (G(D(v))) ∘ E)(u)
= F ((G(D(v)))(E(u))).

Lemma 1.48. The following diagram is commutative.

\xymatrix{Hom(X, Hom(Y, Hom(Z, U))) \ar[r]^{q_{1}} \ar[d]_{Hom(Id_{X},q_{3})} & Hom(X ⊗ Y, Hom(Z, U)) \ar[d]^{q_{2}} \\ Hom(X, Hom(Y ⊗ Z, U)) \ar[r]^{q_{4}} & Hom(X ⊗ Y ⊗ Z, U)}

Proof. For G ∈ Hom(X, Hom(Y, Hom(Z, U))), x ∈ X, y ∈ Y, z ∈ Z,

(q_{2} ∘ q_{1})(G) : x ⊗ y ⊗ z ↦ (q_{2}(q_{1}(G)))(x ⊗ y ⊗ z)
= ((q_{1}(G))(x ⊗ y))(z)
= ((G(x))(y))(z),
(q_{4} ∘ Hom(Id_{X}, q_{3}))(G) : x ⊗ y ⊗ z ↦ (q_{4}(q_{3} ∘ G))(x ⊗ y ⊗ z)
= ((q_{3} ∘ G)(x))(y ⊗ z) = (q_{3}(G(x)))(y ⊗ z)
= ((G(x))(y))(z).

Example 1.49. The generalized transpose map from Definition 1.7 is a distinguished element in the following vector space:

t_{UV}^{W} ∈ Hom(Hom(U, V), Hom(Hom(V, W), Hom(U, W))),

and its image under this q map,

q : Hom(Hom(U, V), Hom(Hom(V, W), Hom(U, W)))
→ Hom(Hom(U, V) ⊗ Hom(V, W), Hom(U, W))

is the following map:

q(t_{UV}^{W}) : Hom(U, V) ⊗ Hom(V, W) → Hom(U, W)
A ⊗ B ↦ (q(t_{UV}^{W}))(A ⊗ B) = (t_{UV}^{W} (A))(B)
(1.5)  = Hom(A, Id_{W})(B) = B ∘ A.

The operation of composition of linear maps is a bilinear function, as in Definition 1.23:

∘ : Hom(U, V) × Hom(V, W) ⇝ Hom(U, W)
(1.6)  (A, B) ↦ B ∘ A.

The agreement of (1.5) and (1.6) shows that q(t_{UV}^{W}) is the unique linear map corresponding to composition as a bilinear function as in (1.4) and Theorem 6.35:

∘ = q(t_{UV}^{W}) ∘ τ .

Remark 1.50. The conclusion of Example 1.49 is that the generalized transpose map is a linear, curried version of composition. Considering

q(t_{UV}^{W}) ∈ Hom(Hom(U, V) ⊗ Hom(V, W), Hom(U, W))

as a distinguished element in a vector space has an analogue in matrix algebra (see, for example, [CHL]), where matrix multiplication can be viewed as an element of a space of tensors.

Example 1.51. Let B : V × V ⇝ 𝕂 be a bilinear function as in Definition 1.23, so that there exists a unique linear map B : V ⊗ V → 𝕂 satisfying B ∘ τ = B as in (1.4). The invertible map q^{−1} : Hom(V ⊗ V, 𝕂) → Hom(V, Hom(V, 𝕂)) transforms B ∈ (V ⊗ V)^{∗} to q^{−1}(B) ∈ Hom(V, V^{∗}). So, every bilinear function B : V × V ⇝ 𝕂 has a linearized, curried form which is an element of Hom(V, V^{∗}), called a \underline{bilinear form} — such maps are the main topic of Chapter 3.

Definition 1.52. For any vector spaces U, V, W define

e_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W) ⊗ U, W)

so that for A : U → V, B : V → W, and u ∈ U,

e_{UV}^{W} (A) : B ⊗ u ↦ B(A(u)) ∈ W.

Lemma 1.53. For any vector spaces U, V, W, U′, V′, W′, and any maps E : U′ → U, F : V → V′, G : W → W′, the following diagram is commutative.

\xymatrix{Hom(U, V) \ar[r]^{e_{UV}^{W}} \ar[dd]_{Hom(E,F)} & Hom(Hom(V, W) ⊗ U, W) \ar[d]^{Hom([Hom(F,Id_{W})⊗E],G)} \\  & Hom(Hom(V′, W) ⊗ U′, W′) \\ Hom(U′, V′) \ar[r]^{e_{U′V ′}^{W′}} & Hom(Hom(V′, W′) ⊗ U′, W′) \ar[u]_{Hom([Hom(Id_{V ′} ,G)⊗Id_{U′}],Id_{W′})}}

Proof. For any A : U → V, C : V′ → W, u ∈ U′,

Hom([Hom(Id_{V′}, G) ⊗ Id_{U′}], Id_{W′}) ∘ e_{U′V′}^{W′} ∘ Hom(E, F) :
A ↦ (e_{U′V′}^{W′}(F ∘ A ∘ E)) ∘ [Hom(Id_{V′}, G) ⊗ Id_{U′}] :
C ⊗ u ↦ (e_{U′V′}^{W′}(F ∘ A ∘ E))((G ∘ C) ⊗ u)
= (G ∘ C)((F ∘ A ∘ E)(u)),
Hom([Hom(F, Id_{W}) ⊗ E], G) ∘ e_{UV}^{W} :
A ↦ G ∘ (e_{UV}^{W} (A)) ∘ [Hom(F, Id_{W}) ⊗ E] :
C ⊗ u ↦ G((e_{UV}^{W} (A))((C ∘ F) ⊗ (E(u))))
= G((C ∘ F)(A(E(u)))).

Lemma 1.54. The following diagram is commutative.

\xymatrix{Hom(U, V) \ar[r]^{e_{UV}^{W}} \ar[d]_{t_{UV}^{W}} & Hom(Hom(V, W) ⊗ U, W) \\ Hom(Hom(V, W), Hom(U, W)) \ar@{-->}[ur]^{q}}

Proof.

q ∘ t_{UV}^{W} : A ↦ q(Hom(A, Id_{W})) : B ⊗ u ↦ (B ∘ A)(u) = (e_{UV}^{W} (A))(B ⊗ u).

Notation 1.55. For the special case W = 𝕂, abbreviate e_{UV}^{𝕂} = e_{UV} (or sometimes just e):

e_{UV} : Hom(U, V) → (V^{∗} ⊗ U)^{∗}

so that for A : U → V, ϕ ∈ V^{∗}, and u ∈ U,

e_{UV} (A) : ϕ ⊗ u ↦ ϕ(A(u)) ∈ 𝕂.

Lemma 1.56. If V is finite-dimensional, then e_{UV} is invertible.

Proof. This follows from Lemma 1.54, Claim 1.11, and Lemma 1.45.

Definition 1.57. For any vector spaces U, V, define

k_{UV} : U^{∗} ⊗ V → Hom(U, V)

so that for ξ ∈ U^{∗}, v ∈ V, and u ∈ U,

(k_{UV} (ξ ⊗ v)) : u ↦ ξ(u) · v ∈ V.

Lemma 1.58. For maps A : U′ → U, B : V → V′, the following diagram is commutative.

\xymatrix{U^{∗} ⊗ V \ar[r]^{k_{UV}} \ar[d]_{[A^{∗}⊗B]} & Hom(U, V) \ar[d]^{Hom(A,B)} \\ U^{′∗} ⊗ V′ \ar[r]^{k_{U′V ′}} & Hom(U′, V′)}

Proof. For ϕ ⊗ v ∈ U^{∗} ⊗ V, u ∈ U′,

(Hom(A, B) ∘ k_{UV})(ϕ ⊗ v) = B ∘ (k_{UV} (ϕ ⊗ v)) ∘ A :
u ↦ B(ϕ(A(u)) · v) = ϕ(A(u)) · B(v),
(k_{U′V′} ∘ [A^{∗} ⊗ B])(ϕ ⊗ v) = k_{U′V′}((A^{∗}(ϕ)) ⊗ (B(v))) :
u ↦ (A^{∗}(ϕ))(u) · B(v) = ϕ(A(u)) · B(v).

Lemma 1.59. For any vector space V, k_{V 𝕂} = l_{V^{∗}} : V^{∗} ⊗ 𝕂 → V^{∗}.

Proof. For v ∈ V, φ ∈ V^{∗}, λ ∈ 𝕂,

k_{V 𝕂} : φ ⊗ λ ↦ k_{V 𝕂}(φ ⊗ λ) : v ↦ φ(v) · λ,
l_{V^{∗}} : φ ⊗ λ ↦ λ · φ : v ↦ (λ · φ)(v).

Lemma 1.60. The canonical map k_{UV} is one-to-one, and if U or V is finitedimensional then k_{UV} is invertible.

Proof. The k_{UV} map (sometimes abbreviated k) is related to a canonical n map. The following diagram is commutative.

\xymatrix{Hom(U, 𝕂) ⊗ V \ar[r]^{n} \ar[dr]_{k_{UV}} & Hom(U, 𝕂 ⊗ V) \ar[d]^{Hom(Id_{U} ,l_{V} )} \\  & Hom(U, V)}

l_{V} ∘ (n(ϕ ⊗ v)) : u ↦ ϕ(u) · v = (k_{UV} (ϕ ⊗ v))(u).

The n map is one-to-one, and, if U or V is finite-dimensional, then n is invertible by Lemma 1.43, which used Claim 1.34. The l_{V} map is invertible as in Example 1.28. See also [B] §II.7.7 or [K] §II.2.

Lemma 1.61. For any vector spaces U, V, W, this diagram is commutative

\xymatrix{ & Hom(U, V) ⊗ W \ar[d]^{n} \\ U^{∗} ⊗ V ⊗ W \ar[ur]^{[k_{UV} ⊗Id_{W}]} \ar[r]_{k_{U,V ⊗W}} & Hom(U, V ⊗ W)}

Proof. For ϕ ⊗ v ⊗ w ∈ U^{∗} ⊗ V ⊗ W and u ∈ U,

ϕ ⊗ v ⊗ w ↦ n((k_{UV} (ϕ ⊗ v)) ⊗ w) : u ↦ (ϕ(u) · v) ⊗ w
ϕ ⊗ v ⊗ w ↦ k_{V,V ⊗W} (ϕ ⊗ v ⊗ w) : u ↦ ϕ(u) · (v ⊗ w).

The diagram can be modified by replacing the n map with a re-ordered version as in Notation 1.40 and adding an arrow to compose with a switching map as in Example 1.29, and is still commutative by a similar calculation.

Remark 1.62. Unlike the canonical maps t_{UV}^{W}, d_{V W}, s, j, n, q, and e_{UV}^{W}, the k maps explicitly refer to the set of scalars 𝕂, in both the dual space U^{∗} and the scalar multiplication · in V. The maps l and m also refer to scalar multiplication.

Remark 1.63. The canonical maps appearing in this Chapter are well-known in abstract linear algebra. Lemma 1.8, Lemma 1.14, Lemma 1.21, Lemma 1.37, Lemma 1.38, Lemma 1.41, Lemma 1.47, Lemma 1.53, and Lemma 1.58 can be interpreted as statements about the naturality of the t, d, m, j, l, n, q, e, and k maps, in a technical sense of category theory. In geometry, these same lemmas also are enough to show that these maps transform in the right way under (pointwise linear, invertible) changes from one local trivialization to another in a vector bundle, so that these basis-free constructions on vector spaces extend to well-defined maps of vector bundles.

Remark 1.64. These canonical maps also appear in concrete matrix algebra and applications: see [Magnus], [G_{2}] (particularly §I.8 and §VI.3), [Graham]; for historical references, see [HS] and [HJ] (Chapter 4). For example, the map k_{UV}^{−1} : Hom(U, V) → U^{∗} ⊗ V is a “vectorization,” or “vec” operator. The t_{UV} map, of course, is analogous to the transpose operation (A ↦ A′, in the notation of [HS] and [Magnus]) and an analogue of Lemma 1.58 is the equality:

vec(ABC) = (C′ ⊗ A)vecB,

attributed by [HS] to [R], see also [Nissen], [HJ], [Magnus] (§1.10).

Notation 1.65. The composite of the e and k maps is denoted

f_{UV} = e_{UV} ∘ k_{UV} : U^{∗} ⊗ V → (V^{∗} ⊗ U)^{∗}.

Sometimes f_{UV} is abbreviated f.

The output f_{UV} (ϕ ⊗ v) acts on ξ ⊗ u to give ϕ(u) · ξ(v) ∈ 𝕂:

(e_{UV} (k_{UV} (ϕ ⊗ v)))(ξ ⊗ u) = ξ((k_{UV} (ϕ ⊗ v))(u)) = ξ(ϕ(u) · v) = ϕ(u) · ξ(v).

Exercise 1.66. For maps A : U′ → U, B : V → V′, the following diagram is commutative.

\xymatrix{U^{∗} ⊗ V \ar[r]^{f_{UV}} \ar[d]_{[A^{∗}⊗B]} & (V^{∗} ⊗ U)^{∗} \ar[d]^{[B^{∗}⊗A]^{∗}} \\ U^{′∗} ⊗ V′ \ar[r]^{f_{U′V ′}} & (V^{′∗} ⊗ U′)^{∗}}

Hint. This can be checked directly as in the Proof of Lemma 1.58; it also follows as a corollary of Lemma 1.53 and Lemma 1.58.

Lemma 1.67. f_{UV}^{∗} ∘ d_{V^{∗}⊗U} = f_{V U} : V^{∗} ⊗ U → (U^{∗} ⊗ V)^{∗}.

Proof. For ξ ⊗ u ∈ V^{∗} ⊗ U, and ϕ ⊗ v ∈ U^{∗} ⊗ V,

(f_{UV}^{∗} (d_{V^{∗}⊗U}(ξ ⊗ u)))(ϕ ⊗ v) = (d_{V^{∗}⊗U}(ξ ⊗ u))(f_{UV} (ϕ ⊗ v))
= (f_{UV} (ϕ ⊗ v))(ξ ⊗ u)
= ϕ(u) · ξ(v)
= (f_{V U}(ξ ⊗ u))(ϕ ⊗ v).

Notation 1.68. For any vector spaces U, V, the following composite is denoted:

p_{UV} = [d_{V} ⊗ Id_{U^{∗}}] ∘ s : U^{∗} ⊗ V → V^{∗∗} ⊗ U^{∗}.

So, for ϕ ∈ U^{∗}, v ∈ V, p_{UV} (ϕ ⊗ v) = (d_{V} (v)) ⊗ ϕ. Sometimes p_{UV} is abbreviated p.

Lemma 1.69. For maps B : V → V′, C : U^{∗} → U^{′∗}, the following diagram is commutative.

\xymatrix{U^{∗} ⊗ V \ar[r]^{p_{UV}} \ar[d]_{[C⊗B]} & V^{∗∗} ⊗ U^{∗} \ar[d]^{[B^{∗∗}⊗C]} \\ U^{′∗} ⊗ V′ \ar[r]^{p_{U′V ′}} & V^{′∗∗} ⊗ U^{′∗}}

Hint. This can be checked using case (1.2) of Lemma 1.14.

Lemma 1.70. The following diagram is commutative. If V is finite-dimensional then all the maps are invertible.

\xymatrix{U^{∗} ⊗ V \ar[r]^{f_{UV}} \ar[d]^{p_{UV}} & (V^{∗} ⊗ U)^{∗} \\ V^{∗∗} ⊗ U^{∗} \ar[r]^{j} & Hom(V^{∗} ⊗ U, 𝕂 ⊗ 𝕂) \ar[u]_{Hom(Id_{V ^{∗}⊗U} ,l)}}

Proof.

ϕ ⊗ v ↦ (Hom(Id_{V^{∗}⊗U}, l) ∘ j ∘ p_{UV})(ϕ ⊗ v) = l ∘ [(d_{V} (v)) ⊗ ϕ] :
ξ ⊗ u ↦ ξ(v) · ϕ(u)
= (f(ϕ ⊗ v))(ξ ⊗ u).

Lemma 1.71. The following diagram is commutative. If V is finite-dimensional then all the maps are invertible.

\xymatrix{U^{∗} ⊗ V \ar[r]^{k_{UV}} \ar[d]_{p_{UV}} & Hom(U, V) \ar[d]^{t_{UV}} \\ V^{∗∗} ⊗ U^{∗} \ar[r]^{k_{V ^{∗}U^{∗}}} & Hom(V^{∗}, U^{∗})}

Proof.

ϕ ⊗ v ↦ (k_{V^{∗}U^{∗}} ∘ p_{UV})(ϕ ⊗ v) = k_{V^{∗}U^{∗}}((d_{V} (v)) ⊗ ϕ) :
ξ ↦ (k_{V^{∗}U^{∗}}((d_{V} (v)) ⊗ ϕ))(ξ) = (d_{V} (v))(ξ) · ϕ = ξ(v) · ϕ :
u ↦ ξ(v) · ϕ(u),
ϕ ⊗ v ↦ (t_{UV} ∘ k_{UV})(ϕ ⊗ v) :
ξ ↦ (t_{UV} (k_{UV} (ϕ ⊗ v)))(ξ) = ξ ∘ (k_{UV} (ϕ ⊗ v)) :
u ↦ ξ((k_{UV} (ϕ ⊗ v))(u)) = ξ(ϕ(u) · v) = ϕ(u) · ξ(v).

The diagram shows how Claim 1.11, Claim 1.16, and Lemma 1.60 are related.

Remark 1.72. The map p corresponds to another well-known object in matrix algebra, denoted the “vec-permutation matrix” I_{m,n} by [HS]. The equality vecA = IvecA′ (in the notation from Remark 1.64) corresponds to the equality k ∘ p = t ∘ k from Lemma 1.71, and [HS] states a matrix analogue of Lemma 1.69. This matrix has also been called the “commutation matrix” K_{mn} by [Magnus] (§3.1) (with the equivalent property KvecA = vecA′) or the “shuffle matrix” by [L], [HJ].

1.3. Direct sums

The following definition applies to any integer ν ≥ 2 (see also [AF] §6).

Definition 1.73. Given vector spaces V, V_{1}, V_{2}, …, V_{ν} and ordered ν-tuples of maps (P_{1}, P_{2}, …, P_{ν}) and (Q_{1}, Q_{2}, …, Q_{ν}), where P_{i} : V → V_{i}, Q_{i} : V_{i} → V for i = 1, 2, …, ν, V is a \underline{direct sum} of V_{1}, V_{2}, …, V_{ν} means:

Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2} + ⋯ + Q_{ν} ∘ P_{ν} = Id_{V}

and

P_{i} ∘ Q_{I} = \{ \begin{matrix}Id_{V_{i}} & if i = I \\ 0_{Hom(V_{I},V_{i})} & if i ≠ I \end{matrix} .

This data is sometimes abbreviated V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{ν}, when the maps P_{i} (called \underline{projections} or \underline{projection operators}) and Q_{i} (\underline{inclusions} or \underline{inclusion operators}) are understood. Note that each map Q_{i} is one-to-one (having a left inverse as in Exercise 0.43) and that if V is finite-dimensional then each V_{i} is also finite-dimensional (by Exercise 0.44 — this fact may not be mentioned at every subsequent occurence).

We will most frequently consider direct sums of two spaces; in this ν = 2 case, the above Definition requires that five equations are satisfied by (P_{1}, P_{2}), (Q_{1}, Q_{2}), but it is enough to check only three of these equations.

Lemma 1.74. Given vector spaces V, V_{1}, V_{2}, if there exist P_{1} : V → V_{1}, P_{2} : V → V_{2}, Q_{1} : V_{1} → V, Q_{2} : V_{2} → V such that:

Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2} = Id_{V}
P_{1} ∘ Q_{1} = Id_{V_{1}}
P_{2} ∘ Q_{2} = Id_{V_{2}}

then V = V_{1} ⊕ V_{2}.

Proof. The conclusion is that the pairs (P_{1}, P_{2}) and (Q_{1}, Q_{2}) satisfy Definition 1.73, and the only two equations remaining to be checked are P_{2} ∘ Q_{1} = 0_{Hom(V_{1},V_{2})} and P_{1} ∘ Q_{2} = 0_{Hom(V_{2},V_{1})}.

P_{2} ∘ Q_{1} = P_{2} ∘ Id_{V} ∘ Q_{1} = P_{2} ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ Q_{1}
= P_{2} ∘ Q_{1} ∘ P_{1} ∘ Q_{1} + P_{2} ∘ Q_{2} ∘ P_{2} ∘ Q_{1}
= P_{2} ∘ Q_{1} ∘ Id_{V_{1}} + Id_{V_{2}} ∘ P_{2} ∘ Q_{1}
= P_{2} ∘ Q_{1} + P_{2} ∘ Q_{1}
= 0_{Hom(V_{1},V_{2})},

the last step using Theorem 0.3. The other equation is similarly checked.

Theorem 1.75. Given vector spaces V, V_{1}, V_{2}, V_{3}, V_{4}, if V = V_{1} ⊕ V_{2} and V_{2} = V_{3} ⊕ V_{4}, then V = V_{1} ⊕ V_{3} ⊕ V_{4}.

Proof. Let (P_{1}, P_{2}), (Q_{1}, Q_{2}) be the operators for V = V_{1} ⊕ V_{2}, and let (P_{3}, P_{4}), (Q_{3}, Q_{4}) be the operators for V_{2} = V_{3} ⊕ V_{4}. The projection operators (P_{1}, P_{3} ∘ P_{2}, P_{4} ∘ P_{2}) and inclusions (Q_{1}, Q_{2} ∘ Q_{3}, Q_{2} ∘ Q_{4}) give a canonical construction for the claimed direct sum. The first equation from Definition 1.73 is:

Q_{1} ∘ P_{1} + (Q_{2} ∘ Q_{3}) ∘ (P_{3} ∘ P_{2}) + (Q_{2} ∘ Q_{4}) ∘ (P_{4} ∘ P_{2})
= Q_{1} ∘ P_{1} + Q_{2} ∘ (Q_{3} ∘ P_{3} + Q_{4} ∘ P_{4}) ∘ P_{2}
= Q_{1} ∘ P_{1} + Q_{2} ∘ Id_{V_{2}} ∘ P_{2}
= Id_{V} .

The remaining nine equations are also easily checked.

Example 1.76. If H : U → V is an invertible map between arbitrary vector spaces, and V = V_{1} ⊕ V_{2}, then U is a direct sum of V_{1} and V_{2}, with projections P_{i} ∘ H : U → V_{i} for i = 1, 2, and inclusions H^{−1} ∘ Q_{i} : V_{i} → U.

Example 1.77. If V = V_{1} ⊕ V_{2}, and U is any vector space, then V ⊗ U is a direct sum of V_{1} ⊗ U and V_{2} ⊗ U. The projection and inclusion operators are [P_{i} ⊗ Id_{U}] : V ⊗ U → V_{i} ⊗ U, and [Q_{i} ⊗ Id_{U}] : V_{i} ⊗ U → V ⊗ U. There is an analogous direct sum U ⊗ V = U ⊗ V_{1} ⊕ U ⊗ V_{2}.

Example 1.78. If V = V_{1} ⊕V_{2}, and U is any vector space, then Hom(U, V) is a direct sum of Hom(U, V_{1}) and Hom(U, V_{2}). The projection and inclusion operators are Hom(Id_{U}, P_{i}) : Hom(U, V) → Hom(U, V_{i}), and Hom(Id_{U}, Q_{i}) : Hom(U, V_{i}) → Hom(U, V).

Example 1.79. If V = V_{1} ⊕ V_{2}, and U is any vector space, then Hom(V, U) is a direct sum of Hom(V_{1}, U) and Hom(V_{2}, U). The projection operators are Hom(Q_{i}, Id_{U}) : Hom(V, U) → Hom(V_{i}, U), and the inclusion operators are

Hom(P_{i}, Id_{U}) : Hom(V_{i}, U) → Hom(V, U).

Example 1.80. As the U = 𝕂 special case of the previous Example, if V = V_{1} ⊕ V_{2}, then V^{∗} = V_{1}^{∗} ⊕ V_{2}^{∗}, with projections Q_{i}^{∗} and inclusions P_{i}^{∗}.

Lemma 1.81. Given V = V_{1} ⊕ V_{2}, the image of Q_{2}, i.e., the subspace Q_{2}(V_{2}) of V, is equal to the subspace ker(P_{1}), the kernel of P_{1}, which is also equal to ker(Q_{1} ∘ P_{1}).

Proof. The second equality follows from ker(P_{1}) ⊆ ker(Q_{1}∘P_{1}) ⊆ ker(P_{1}∘Q_{1}∘ P_{1}) = ker(P_{1}). It follows from P_{1} ∘ Q_{2} = 0_{Hom(V_{2},V_{1})} that Q_{2}(V_{2}) ⊆ ker(P_{1}), and if P_{1}(v) = 0_{V_{1}}, then v = (Q_{1} ∘P_{1} +Q_{2} ∘P_{2})(v) = Q_{2}(P_{2}(v)), so ker(P_{1}) ⊆ Q_{2}(V_{2}).

Lemma 1.82. Given U = U_{1} ⊕ U_{2} and V = V_{1} ⊕ V_{2}, with projections and inclusions P′_{i}, Q′_{i}, P_{i}, Q_{i}, respectively, if the maps A_{1} : U_{1} → V_{1} and A_{2} : U_{2} → V_{2} are both invertible, then the map Q_{1} ∘ A_{1} ∘ P′_{1} + Q_{2} ∘ A_{2} ∘ P′_{2} : U → V is invertible.

Proof. The inverse is Q′_{1} ∘ A_{1}^{−1} ∘ P_{1} + Q′_{2} ∘ A_{2}^{−1} ∘ P_{2}.

Lemma 1.83. Given a direct sum V = V_{1} ⊕ V_{2} as in Definition 1.73, another direct sum U = U_{1} ⊕ U_{2}, with operators P′_{i} and Q′_{i}, and H : U → V, the following are equivalent.

(1) Q_{1} ∘ P_{1} ∘ H = H ∘ Q′_{1} ∘ P′_{1}.

(2) Q_{2} ∘ P_{2} ∘ H = H ∘ Q′_{2} ∘ P′_{2}.

(3) P_{1} ∘ H ∘ Q′_{2} = 0_{Hom(U_{2},V_{1})} and P_{2} ∘ H ∘ Q′_{1} = 0_{Hom(U_{1},V_{2})}.

(4) There exist maps H_{1} : U_{1} → V_{1} and H_{2} : U_{2} → V_{2} such that H =

Q_{1} ∘ H_{1} ∘ P′_{1} + Q_{2} ∘ H_{2} ∘ P′_{2}.

Proof. First, for (1) ⟺ (2),

H = (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ H = H ∘ (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2})
= Q_{1} ∘ P_{1} ∘ H + Q_{2} ∘ P_{2} ∘ H = H ∘ Q′_{1} ∘ P′_{1} + H ∘ Q′_{2} ∘ P′_{2}.

Applying either equality (1) or (2), then subtracting, gives the other equality. For (1) ⟹ (3),

P_{1} ∘ H ∘ Q′_{2} = P_{1} ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ H ∘ Q′_{2}
= P_{1} ∘ H ∘ Q′_{1} ∘ P′_{1} ∘ Q′_{2} = 0_{Hom(U_{2},V_{1})},
P_{2} ∘ H ∘ Q′_{1} = P_{2} ∘ H ∘ (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2}) ∘ Q′_{1}
= P_{2} ∘ Q_{1} ∘ P_{1} ∘ H ∘ Q′_{1} = 0_{Hom(U_{1},V_{2})}.

Next, to show that (3) implies (1) or (2), let i = 1 or 2:

Q_{i} ∘ P_{i} ∘ H = Q_{i} ∘ P_{i} ∘ H ∘ (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2})
= Q_{i} ∘ P_{i} ∘ H ∘ Q′_{i} ∘ P′_{i}
= (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ H ∘ Q′_{i} ∘ P′_{i} = H ∘ Q′_{i} ∘ P′_{i}.

The construction in (4) is the same as in Lemma 1.82 (without requiring invertibility). The implication (4) ⟹ (1) is straightforward. To show that (1) and (2) imply (4), let H_{1} = P_{1} ∘ H ∘ Q′_{1} and H_{2} = P_{2} ∘ H ∘ Q′_{2}. Then,

Q_{1} ∘ H_{1} ∘ P′_{1} + Q_{2} ∘ H_{2} ∘ P′_{2} = Q_{1} ∘ P_{1} ∘ H ∘ Q′_{1} ∘ P′_{1} + Q_{2} ∘ P_{2} ∘ H ∘ Q′_{2} ∘ P′_{2}
= Q_{1} ∘ P_{1} ∘ Q_{1} ∘ P_{1} ∘ H + Q_{2} ∘ P_{2} ∘ Q_{2} ∘ P_{2} ∘ H
= (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ H = H.

Definition 1.84. For V = V_{1} ⊕ V_{2}, U = U_{1} ⊕ U_{2}, and H : U → V, H \underline{respects the direct sums} means: H satisfies any of the equivalent conditions from Lemma 1.83. For such a map and i = 1, 2, the composites P_{i} ∘ H ∘ Q′_{i} : U_{i} → V_{i} are said to be induced by H.

Lemma 1.85. If H : U → V is an invertible map which respects the direct sums as in Definition 1.84, then H^{−1} also respects the direct sums, and for each i = 1, 2, the induced map P_{i} ∘ H ∘ Q′_{i} : U_{i} → V_{i} is invertible, with inverse P′_{i} ∘ H^{−1} ∘ Q_{i}.

Proof.

(P_{i} ∘ H ∘ Q′_{i}) ∘ (P′_{i} ∘ H^{−1} ∘ Q_{i}) = P_{i} ∘ Q_{i} ∘ P_{i} ∘ H ∘ H^{−1} ∘ Q_{i}
= P_{i} ∘ Q_{i} = Id_{V_{i}}
(P′_{I} ∘ H^{−1} ∘ Q_{i}) ∘ (P_{i} ∘ H ∘ Q′_{i}) = P′_{I} ∘ H^{−1} ∘ H ∘ Q′_{i} ∘ P′_{i} ∘ Q′_{i}
= P′_{I} ∘ Q′_{i}

If i = I, this shows P_{i}∘H∘Q′_{i} is invertible. If i ≠ I, then P′_{I}∘H^{−1}∘Q_{i} = 0_{Hom(V_{i},U_{I})}, so H^{−1} respects the direct sums.

Lemma 1.86. Given U = U_{1} ⊕ U_{2}, V = V_{1} ⊕ V_{2}, W = W_{1} ⊕ W_{2}, if H : U → V respects the direct sums and H′ : V → W respects the direct sums, then H′ ∘ H : U → W respects the direct sums. A map induced by the composite is equal to the composite of the corresponding induced maps.

Lemma 1.87. Suppose V = V_{1} ⊕ V_{2}, U = U_{1} ⊕ U_{2}, and H : U → V respects the direct sums, inducing maps P_{i} ∘ H ∘ Q′_{i}. Then, for any A : W → X, the map [H ⊗ A] : U ⊗ W → V ⊗ X respects the direct sums

U_{1} ⊗ W ⊕ U_{2} ⊗ W → V_{1} ⊗ X ⊕ V_{2} ⊗ X

from Example 1.77. The induced map [P_{i} ⊗ Id_{X}] ∘ [H ⊗ A] ∘ [Q′_{i} ⊗ Id_{W}] is equal to [(P_{i} ∘ H ∘ Q′_{i}) ⊗ A].

Proof. All the claims follow immediately from Lemma 1.36.

Lemma 1.88. Suppose V = V_{1} ⊕ V_{2}, U = U_{1} ⊕ U_{2}, and H : U → V respects the direct sums, inducing maps P_{i} ∘ H ∘ Q′_{i}. Then, for any A : W → X, the map Hom(A, H) : Hom(X, U) → Hom(W, V) respects the direct sums

Hom(X, U_{1}) ⊕ Hom(X, U_{2}) → Hom(W, V_{1}) ⊕ Hom(W, V_{2})

from Example 1.78. The induced map Hom(Id_{W} , P_{i}) ∘ Hom(A, H) ∘ Hom(Id_{X}, Q′_{i}) is equal to Hom(A, P_{i} ∘ H ∘ Q′_{i}). Analogously, the map Hom(H, A) : Hom(V, W) → Hom(U, X) respects the direct sums

Hom(V_{1}, W) ⊕ Hom(V_{2}, W) → Hom(U_{1}, X) ⊕ Hom(U_{2}, X)

from Example 1.79, and the induced map Hom(Q′_{i}, Id_{W})∘Hom(H, A)∘Hom(P_{i}, Id_{X}) is equal to Hom(P_{i} ∘ H ∘ Q′_{i}, A).

Proof. All the claims follow immediately from Lemma 1.6.

Notation 1.89. Given a direct sum V = V_{1} ⊕ V_{2} with operator pairs (P_{1}, P_{2}), (Q_{1}, Q_{2}), the pairs in the other order, (P_{2}, P_{1}), (Q_{2}, Q_{1}), also satisfy the definition of direct sum. The notation V = V_{2} ⊕ V_{1} refers to these re-ordered pairs.

Example 1.90. A map H ∈ End(V) which satisfies Q_{2} ∘ P_{2} ∘ H = H ∘ Q_{1} ∘ P_{1} respects the direct sums H : V_{1} ⊕ V_{2} → V_{2} ⊕ V_{1} (and so H also satisfies the other identities from Lemma 1.83).

Lemma 1.91. Given a vector space V that admits two direct sums, V = V_{1}⊕V_{2}, V = V′′_{1} ⊕ V′′_{2} with projections and inclusions P_{i}, Q_{i}, P′′_{i}, Q′′_{i}, respectively, the following are equivalent.

(1) The identity map Id_{V} : V_{1} ⊕ V_{2} → V′′_{1} ⊕ V′′_{2} respects the direct sums.

(2) Q_{1} ∘ P_{1} = Q′′_{1} ∘ P′′_{1}.

(3) Q_{2} ∘ P_{2} = Q′′_{2} ∘ P′′_{2}.

(4) P′′_{I} ∘ Q_{i} = 0_{Hom(V_{i},V′′_{I})} for i ≠ I.

(5) P_{I} ∘ Q′′_{i} = 0_{Hom(V′′_{i},V_{I})} for i ≠ I.

Proof. The first statement is, by Definition 1.84 and Lemma 1.83, equivalent to any of the next three statements. The equivalence with the last statement follows from Lemma 1.85.

Definition 1.92. Given V, two direct sums V = V_{1} ⊕ V_{2} and V = V′′_{1} ⊕ V′′_{2} are \underline{equivalent direct sums} means: they satisfy any of the properties from Lemma 1.91.

For a fixed V, this notion is clearly an equivalence relation on direct sum decompositions of V.

Example 1.93. If V = V_{1} ⊕ V_{2} and H_{1} : U_{1} → V_{1} and H_{2} : U_{2} → V_{2} are invertible, then V is a direct sum U_{1} ⊕ U_{2}, with projections H_{i}^{−1} ∘ P_{i} and inclusions Q_{i} ∘ H_{i}. The direct sums V = V_{1} ⊕ V_{2} and V = U_{1} ⊕ U_{2} are equivalent.

Lemma 1.94. Given U and V, a direct sum U = U_{1} ⊕ U_{2}, and a map H : U → V, suppose V = V_{1} ⊕ V_{2} and V = V′′_{1} ⊕ V′′_{2} are equivalent direct sums. Then H respects the direct sums U_{1} ⊕ U_{2} → V_{1} ⊕ V_{2} if and only if H respects the direct sums U_{1} ⊕ U_{2} → V′′_{1} ⊕ V′′_{2}. Similarly, a map A : V → U respects the direct sums V_{1} ⊕ V_{2} → U_{1} ⊕ U_{2} if and only if A respects the direct sums V′′_{1} ⊕ V′′_{2} → U_{1} ⊕ U_{2}.

Lemma 1.95. Given V = V_{1} ⊕ V_{2} and V = V′′_{1} ⊕ V′′_{2} with operators P_{i}, Q_{i}, P′′_{i}, Q′′_{i}, if P_{1} ∘ Q′′_{2} = 0_{Hom(V′′_{2},V_{1})}, and P′′_{1} ∘ Q_{2} = 0_{Hom(V_{2},V′′_{1})}, then P′′_{1} ∘ Q_{1} : V_{1} → V′′_{1} and P′′_{2} ∘ Q_{2} : V_{2} → V′′_{2} are both invertible.

Proof. The inverse of P′′_{i} ∘ Q_{i} is P_{i} ∘ Q′′_{i} : V′′_{i} → V_{i}.

As a special case of both Lemma 1.95 and Lemma 1.85, if V = V_{1} ⊕V_{2} and V = V′′_{1} ⊕ V′′_{2} are equivalent direct sums, then there are canonically induced invertible maps P′′_{i} ∘ Q_{i} : V_{i} → V′′_{i}, i = 1, 2.

Lemma 1.96. Suppose ϕ ∈ V^{∗}. If ϕ ≠ 0_{V^{∗}} then there exists a direct sum V = 𝕂 ⊕ ker(ϕ).

Proof. Let Q_{2} be the inclusion of the kernel subspace ker(ϕ) = {w ∈ V : ϕ(w) = 0} in V. Since ϕ ≠ 0_{V^{∗}}, there exists some v ∈ V so that ϕ(v) ≠ 0. Let α, β ∈ 𝕂 be any constants so that α · β · ϕ(v) = 1. Define Q_{1}^{β} : 𝕂 → V so that for γ ∈ 𝕂, Q_{1}^{β}(γ) = β · γ · v. Define P_{1}^{α} = α · ϕ : V → 𝕂. Then,

P_{1}^{α} ∘ Q_{1}^{β} : γ ↦ α · ϕ(β · γ · v) = α · β · ϕ(v) · γ = γ.

For any w ∈ ker(ϕ), (P_{1}^{α} ∘ Q_{2})(w) = α · ϕ(w) = 0. Define P_{2} = Id_{V} − Q_{1}^{β} ∘ P_{1}^{α}, which is a map from V to ker(ϕ): if u ∈ V, then

(ϕ ∘ P_{2})(u) = (ϕ ∘ Id_{V} − ϕ ∘ Q_{1}^{β} ∘ P_{1}^{α})(u)
= ϕ(u) − ϕ(Q_{1}^{β}(α · ϕ(u)))
= ϕ(u) − ϕ(β · α · ϕ(u) · v)
= ϕ(u) − α · β · ϕ(v) · ϕ(u) = 0.

Also, for w ∈ ker(ϕ),

(P_{2} ∘ Q_{2})(w) = ((Id_{V} − Q_{1}^{β} ∘ P_{1}^{α}) ∘ Q_{2})(w) = (Q_{2} − Q_{1}^{β} ∘ 0_{(ker(ϕ))^{∗}})(w) = w,

so P_{2} ∘ Q_{2} = Id_{ker(ϕ)}, and the claim follows from Lemma 1.74.

Given V and ϕ, the direct sum from the previous Lemma is generally not unique, nor are two such direct sums, depending on v, α, β, even equivalent in general. However, in some later examples, there will be a canonical element v ∉ ker(ϕ), and in such a case, different choices of α, β give equivalent direct sums.

Lemma 1.97. Given V, ϕ ∈ V^{∗}, and v ∈ V so that ϕ(v) ≠ 0, let α, β, α′, β′ ∈ 𝕂 be any constants so that α · β · ϕ(v) = α′ · β′ · ϕ(v) = 1. Then the direct sum V = 𝕂 ⊕ ker(ϕ) constructed in the Proof of Lemma 1.96 is equivalent to the analogous direct sum with operators Q_{1}^{β′} : γ ↦ β′ · γ · v, P_{1}^{α′} = α′ · ϕ, Q_{2}, and P′_{2} = Id_{V} − Q_{1}^{β′} ∘ P_{1}^{α′}.

The following result will be used as a step in Theorem 4.40.

Theorem 1.98. Suppose U = U_{1} ⊕ U_{2} is a direct sum with projection and inclusion operators P_{i}, Q_{i}, and that there are vector spaces V, V_{1}, V_{2}, and maps P′_{1} : V → V_{1}, Q′_{2} : V_{2} → V, H : U → V, H_{1} : U_{1} → V_{1}, H_{2} : U_{2} → V_{2}, such that P′_{1} ∘ H = H_{1} ∘ P_{1}, Q′_{2} ∘ H_{2} = H ∘ Q_{2}, and P′_{1} ∘ Q′_{2} = 0_{Hom(V_{2},V_{1})}. Suppose further that H and H_{1} are invertible, and that Q′_{2} is a linear monomorphism. Then, there exist maps Q′_{1} : V_{1} → V and P′_{2} : V → V_{2} such that V = V_{1} ⊕ V_{2}. Also, H respects the direct sums, and H_{2} is invertible.

Proof. Let Q′_{1} = H ∘ Q_{1} ∘ H_{1}^{−1}. Then
P′_{1} ∘ Q′_{1} = P′_{1} ∘ H ∘ Q_{1} ∘ H_{1}^{−1} = H_{1} ∘ P_{1} ∘ Q_{1} ∘ H_{1}^{−1} = Id_{V_{1}}.

Let P′_{2} = H_{2} ∘ P_{2} ∘ H^{−1}. Then

Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2} = H ∘ Q_{1} ∘ H_{1}^{−1} ∘ P′_{1} + Q′_{2} ∘ H_{2} ∘ P_{2} ∘ H^{−1}
= H ∘ Q_{1} ∘ P_{1} ∘ H^{−1} + H ∘ Q_{2} ∘ P_{2} ∘ H^{−1}
= Id_{V} .
Q′_{2} ∘ P′_{2} ∘ Q′_{2} = (Id_{V} − Q′_{1} ∘ P′_{1}) ∘ Q′_{2} = Q′_{2},

so P′_{2} ∘Q′_{2} = Id_{V_{2}}, by the monomorphism property (Definition 0.42), and this shows V = V_{1} ⊕ V_{2} by Lemma 1.74. H respects the direct sums:

P′_{1} ∘ H ∘ Q_{2} = H_{1} ∘ P_{1} ∘ Q_{2} = 0_{Hom(U_{2},V_{1})}
P′_{2} ∘ H ∘ Q_{1} = H_{2} ∘ P_{2} ∘ H^{−1} ∘ H ∘ Q_{1} = 0_{Hom(U_{1},V_{2})}.

By Lemma 1.85, P′_{2} ∘ H ∘ Q_{2} = H_{2} ∘ P_{2} ∘ H^{−1} ∘ H ∘ Q_{2} = H_{2} has inverse P_{2} ∘ H^{−1} ∘ Q′_{2}.

Exercise 1.99. Let V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{ν} be a direct sum, as in Definition 1.73, with projections (P_{1}, P_{2}, …, P_{ν}) and inclusions (Q_{1}, Q_{2}, …, Q_{ν}). For another vector space W and additive functions A : V ⇝ W, B : V ⇝ W, the following are equivalent.

(1) For all i = 1, …, ν, A ∘ Q_{i} = B ∘ Q_{i} : V_{i} → W.

(2) A = B.

Hint. The additive property is used in steps (1.7), (1.8).

A = A ∘ (Q_{1} ∘ P_{1} + ⋯ + Q_{ν} ∘ P_{ν})
(1.7)  = (A ∘ Q_{1}) ∘ P_{1} + ⋯ + (A ∘ Q_{ν}) ∘ P_{ν}
= (B ∘ Q_{1}) ∘ P_{1} + ⋯ + (B ∘ Q_{ν}) ∘ P_{ν}
(1.8)  = B ∘ (Q_{1} ∘ P_{1} + ⋯ + Q_{ν} ∘ P_{ν})
= B.

Exercise 1.100. Let V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{ν} be a direct sum. Suppose there is some i and some additive function P′ : V ⇝ V_{i} so that P′ ∘ Q_{i} = Id_{V_{i}}, and P′ ∘ Q_{I} = 0_{Hom(V_{I},V_{i})} for I ≠ i. Then, P′ = P_{i}.

Hint. This is a special case of Exercise 1.99. P′ is not assumed to be linear, but the above calculation uses the additive property to conclude that P′ is linear because it equals a given linear map.

In the above sense, given all the operators P_{1}, …, Q_{ν} in a direct sum, each individual map P_{i} is unique. Exercise 1.102 states an analogous uniqueness result for Q_{i}.

Exercise 1.101. Let V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{ν} be a direct sum. For any set W and any functions A : W ⇝ V, B : W ⇝ V, the following are equivalent.

(1) For all i = 1, …, ν, P_{i} ∘ A = P_{i} ∘ B : W → V_{i}.

(2) A = B.

Hint. The calculation is similar to that in Exercise 1.99, but does not assume any additive property.

Exercise 1.102. For V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{ν}, suppose there is some i and some function Q′ : V_{i} ⇝ V so that P_{i} ∘ Q′ = Id_{V_{i}}, and P_{I} ∘ Q′ = 0_{Hom(V_{i},V_{I})} for I ≠ i. Then, Q′ = Q_{i}.

Hint. This is a special case of Exercise 1.101.

Exercise 1.103. Let V = V_{1}⊕V_{2} be a direct sum with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), and let A : V_{1} → V_{2}. Then the following operators (P′_{1}, P′_{2}), (Q′_{1}, Q′_{2}) also define a direct sum.

Q′_{1} = Q_{1} + Q_{2} ∘ A : V_{1} → V
Q′_{2} = Q_{2} : V_{2} → V
P′_{1} = P_{1} : V → V_{1}
P′_{2} = P_{2} − A ∘ P_{1} : V → V_{2}.

This is equivalent to the original direct sum if and only if A = 0_{Hom(V_{1},V_{2})}.

Hint. The first equation from Definition 1.73 is:

Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2} = (Q_{1} + Q_{2} ∘ A) ∘ P_{1} + Q_{2} ∘ (P_{2} − A ∘ P_{1})
= Q_{1} ∘ P_{1} + Q_{2} ∘ A ∘ P_{1} + Q_{2} ∘ P_{2} − Q_{2} ∘ A ∘ P_{1}
= Id_{V} .

The remaining equations from Definition 1.73 (or Lemma 1.74) are also easy to check. If the direct sums are equivalent, then 0_{Hom(V_{1},V_{2})} = P_{2} ∘ Q′_{1} = P_{2} ∘ (Q_{1} + Q_{2} ∘ A) = A, and conversely.

The direct sum P′_{i}, Q′_{i} is the \underline{graph} of A. In a certain sense, Exercise 1.103 has a converse: if a space V decomposes in two ways as a direct sum, with the same inclusion Q_{2}, then the two direct sums are related using the graph construction, up to equivalence.

Exercise 1.104. Given V, V_{1}, V_{2}, suppose the pairs (P_{1}, P_{2}) and (Q_{1}, Q_{2}) define a direct sum V = V_{1} ⊕ V_{2}, and the pairs (P′_{1}, P′_{2}), (Q′_{1}, Q′_{2}) also satisfy Definition 1.73. If Q_{2} = Q′_{2}, then there exists a map A : V_{1} → V_{2}, and there exist (P′′_{1}, P′′_{2}), (Q′′_{1}, Q′′_{2}) which define a third direct sum, and which satisfy:

Q′′_{1} = Q_{1} + Q_{2} ∘ A : V_{1} → V
Q′′_{2} = Q_{2} : V_{2} → V
P′′_{1} = P_{1} : V → V_{1}
P′′_{2} = P_{2} − A ∘ P_{1} : V → V_{2}.

This (P′′_{1}, P′′_{2}), (Q′′_{1}, Q′′_{2}) direct sum is equivalent to the (P′_{1}, P′_{2}), (Q′_{1}, Q′_{2}) direct sum.

Hint. Note that by Exercise 1.102, the hypothesis Q_{2} = Q′_{2} is equivalent to assuming that Q′_{2} satisfies P_{2} ∘ Q′_{2} = Id_{V_{2}} and P_{1} ∘ Q′_{2} = 0_{Hom(V_{2},V_{1})}. Choosing Q′′_{2} = Q_{2} gives the identity P′_{1} ∘ Q′′_{2} = 0_{Hom(V_{2},V_{1})}.

The above four equations are the properties defining a graph. The claimed existence follows from checking that the following choices have the claimed properties.

A = −P′_{2} ∘ Q_{1} : V_{1} → V_{2}
Q′′_{1} = Q′_{1} ∘ P′_{1} ∘ Q_{1} : V_{1} → V
P′′_{2} = P′_{2} : V → V_{2}.

The equivalence of direct sums as in Definition 1.92 is verified by checking P′_{2}∘Q′′_{1} = 0_{Hom(V_{1},V_{2})}.

1.4. Idempotents and involutions

Definition 1.105. An element P ∈ End(V) is an \underline{idempotent} means: P ∘ P = P.

Lemma 1.106. Given V and P_{1}, P_{2} ∈ End(V), any three out of the following four properties (1) − (4) imply the remaining fourth.

(1) P_{1} is an idempotent.

(2) P_{2} is an idempotent.

(3) P_{1} ∘ P_{2} + P_{2} ∘ P_{1} = 0_{End(V)}.

(4) P_{1} + P_{2} is an idempotent. Property (4) is equivalent to:

(5) There exists P_{3} ∈ End(V) such that P_{3} is an idempotent and P_{1}+P_{2}+P_{3} =

Id_{V}.

If, further, either \frac{1}{2} ∈ 𝕂 or P_{1} + P_{2} = Id_{V}, then P_{1}, P_{2}, P_{3} satisfying properties (1) − (5) also satisfy:

(6) For distinct i_{1}, i_{2} ∈ {1, 2, 3}, P_{i_{1}} ∘ P_{i_{2}} = 0_{End(V)}. Conversely, if P_{1}, P_{2}, P_{3} ∈ End(V) satisfy (6) and P_{1} + P_{2} + P_{3} = Id_{V}, then P_{1}, P_{2}, P_{3} are idempotents satisfying (1) − (5).

Exercise 1.107. For P ∈ End(V), the following are equivalent.

(1) P is an idempotent.

(2) For any A ∈ End(V), P + P ∘ A − P ∘ A ∘ P is an idempotent.

(3) For any A ∈ End(V), P + A ∘ P − P ∘ A ∘ P is an idempotent.

Exercise 1.108. If P ∈ End(V) is an idempotent, then the following are equivalent.

(1) P′ ∘ P = P ∘ P′ for all idempotents P′ ∈ End(V).

(2) P = 0_{End(V)} or P = Id_{V}.

Hint. This follows from Exercise 1.107 and Claim 0.51.

Example 1.109. An idempotent P defines a direct sum structure as follows. Let V_{1} = P (V), the subspace of V which is the image of P. Define Q_{1} : V_{1} → V to be the subspace inclusion map, and define P_{1} : V → V_{1} by restricting the target of P : V → V to get P_{1} : V → V_{1} with P_{1}(v) = P (v) for all v ∈ V. Then P = Q_{1} ∘ P_{1} : V → V by construction. The map Id_{V} − P is also an idempotent (this is a special case of Lemma 1.106), so proceeding analogously, define V_{2} = (Id_{V} − P)(V), the image of the linear map Id_{V} − P : V → V. Again, let Q_{2} : V_{2} → V be the subspace inclusion, and define P_{2} = Id_{V} − P, with its target space restricted to V_{2}, so that Q_{2} ∘ P_{2} = Id_{V} − P by construction, and Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2} = P + (Id_{V} − P) = Id_{V}. To show V = V_{1} ⊕ V_{2}, it remains only to check that these maps satisfy the two remaining equations from Lemma 1.74. For v_{1} ∈ V_{1}, Q_{1}(v_{1}) = v_{1} = P (w_{1}) for some w_{1} ∈ V, so (P_{1} ∘ Q_{1})(v_{1}) = P_{1}(P (w_{1})) = P (P (w_{1})) = P (w_{1}) = v_{1}. Similarly, for Q_{2}(v_{2}) = v_{2} = (Id_{V} −P)(w_{2}), (P_{2}∘Q_{2})(v_{2}) = (Id_{V} −P)((Id_{V} −P)(w_{2})) = (Id_{V} −P)(w_{2}) = v_{2}. This construction of the direct sum V = V_{1} ⊕ V_{2} is canonical up to re-ordering.

The statement of Lemma 1.81 in the special case of Example 1.109 is that the image of Id_{V} − P is the kernel of P, and the image of P is the kernel of Id_{V} − P.

Example 1.110. Given any direct sum V = U_{1} ⊕ U_{2} as in Definition 1.73 with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), the composite Q_{1} ∘ P_{1} : V → V is an idempotent, and so is Id_{V} − Q_{1} ∘ P_{1} = Q_{2} ∘ P_{2}. This is a converse to the construction of Example 1.109; any direct sum canonically defines an unordered pair of two idempotents. For P = Q_{1} ∘ P_{1}, the direct sum V = V_{1} ⊕ V_{2} constructed in Example 1.109 is equivalent, as in Definition 1.92, to the original direct sum.

Lemma 1.111. Given idempotents P : V → V, P′ : U → U defining direct sums V_{1} ⊕ V_{2} and U_{1} ⊕ U_{2} as in Example 1.109, and a map H : U → V, the following are equivalent.

(1) H respects the direct sums (as in Definition 1.84).

(2) H ∘ P′ = P ∘ H.

Definition 1.112. An element K ∈ End(V) is an \underline{involution} means: K ∘ K = Id_{V}.

Lemma 1.113. If \frac{1}{2} ∈ 𝕂 and K ∈ End(V), then the following are equivalent.

(1) K ∈ End(V) is an involution.

(2) P = \frac{1}{2} · (Id_{V} + K) is an idempotent.

(3) Id_{V} − P = \frac{1}{2} · (Id_{V} − K) is an idempotent.

Lemma 1.114. For an involution K ∈ End(V), let V_{1} = {v ∈ V : K(v) = v}, and V_{2} = \{v ∈ V : K(v) = −v\}. If \frac{1}{2} ∈ 𝕂, then V = V_{1} ⊕ V_{2}, with Q_{i} the subspace inclusion maps, and projections:

(1.9)  P_{1} = \frac{1}{2} · (Id_{V} + K),
(1.10)  P_{2} = \frac{1}{2} · (Id_{V} − K).

Proof. This can be proved directly, but also follows from the construction of Example 1.109. It is easy to check that V_{1} is a subspace of V, equal to the image of the idempotent P from Lemma 1.113 and that V_{2} is equal to the image of Id_{V} − P. The composites Q_{1} ∘ P_{1}, Q_{2} ∘ P_{2} ∈ End(V) are also given by the formulas \frac{1}{2} · (Id_{V} ± K).

Notation 1.115. We refer to the construction of V = V_{1} ⊕ V_{2} as in Lemma 1.114 as the \underline{direct sum produced by the involution} K. The subspaces V_{1}, V_{2} and maps P_{1}, P_{2} in (1.9), (1.10) are canonical, but Lemma 1.114 made a choice of order in the direct sum V = V_{1} ⊕ V_{2}. With this ordering convention, the involution −K produces the direct sum V = V_{2} ⊕ V_{1} as in Notation 1.89. For the projection maps defined by formulas (1.9), (1.10), the double arrow will appear in diagrams, P_{1} : V ↠ V_{1}, P_{2} : V ↠ V_{2}, and the same arrow for composites of such projections. For the subspace inclusion maps as in Lemma 1.114, the hook arrow will appear: Q_{1} : V_{1} ↪ V for the fixed point subspace of K, and Q_{2} : V_{2} ↪ V for the fixed point subspace of −K, and similarly for composites of such inclusions.

Example 1.116. Given any direct sum V = U_{1} ⊕ U_{2} as in Definition 1.73 with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), the map

Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2} = Id_{V} − 2 · Q_{2} ∘ P_{2} : V → V

is an involution, and it respects the direct sums U_{1} ⊕ U_{2} → U_{1} ⊕ U_{2}. If \frac{1}{2} ∈ 𝕂, then the direct sum produced by this involution, V = V_{1} ⊕ V_{2} as in Lemma 1.114, is equivalent, as in Definition 1.92, to the original direct sum. As in Lemma 1.85 and Lemma 1.95, there are invertible maps U_{i} → V_{i}. If the direct sum operators Q_{i}, P_{i} were defined by some involution K as in Lemma 1.114, then Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2} = K.

Lemma 1.117. For an idempotent P : V → V, let V = V_{1}⊕V_{2} be the direct sum from Example 1.109. The maps K = 2 · P − Id_{V} : V → V and Id_{V} − 2 · P = −K are involutions, and both K and −K respect the direct sums V_{1} ⊕ V_{2} → V_{1} ⊕ V_{2}. If \frac{1}{2} ∈ 𝕂, then the direct sum from Lemma 1.114 produced by K is the same as V = V_{1} ⊕ V_{2}.

Proof. The claim that K and −K respect the direct sums is a special case of Lemma 1.111.

Example 1.118. For any space V, the switching map s : V ⊗ V → V ⊗ V from Example 1.29 is an involution. For \frac{1}{2} ∈ 𝕂, s produces a direct sum on V ⊗ V, denoted:

V ⊗ V = S^{2}V ⊕ Λ^{2}V,

with projections P_{1} = \frac{1}{2} · (Id_{V ⊗V} + s) : V ⊗ V ↠ S^{2}V and P_{2} = \frac{1}{2} · (Id_{V ⊗V} − s) : V ⊗ V ↠ Λ^{2}V and corresponding subspace inclusions Q_{1}, Q_{2}.

Theorem 1.119. Given \frac{1}{2} ∈ 𝕂, an involution K ∈ End(V), let V = V_{1} ⊕ V_{2} be the direct sum produced by K. For any ϕ ∈ V^{∗}, if ϕ ≠ 0_{V^{∗}} and ϕ ∘ K = ϕ, then there is a direct sum V = 𝕂 ⊕ ker(ϕ ∘ Q_{1}) ⊕ V_{2}.

Proof. Let (P_{1}, P_{2}) and (Q_{1}, Q_{2}) be the operators as in Lemma 1.114; the inclusion Q_{1} appears in the claim. From ϕ ≠ 0_{V^{∗}}, there is some w ∈ V so that ϕ(w) ≠ 0. Let v = \frac{1}{2} · (w + K(w)) ∈ V_{1}. Then

(ϕ ∘ Q_{1})(v) = ϕ(v) = ϕ(\frac{1}{2} · (w + K(w))) = \frac{1}{2} · ϕ(w) + \frac{1}{2} · ϕ(K(w)) = ϕ(w) ≠ 0.

So ϕ ∘ Q_{1} ≠ 0_{V_{1}^{∗}} and Lemma 1.96 applies to get a direct sum V_{1} = 𝕂 ⊕ ker(ϕ ∘ Q_{1}), depending on parameters α, β ∈ 𝕂 such that α · β · ϕ(w) = 1. The inclusions are Q_{3}^{β} : 𝕂 → V_{1} : γ ↦ β · γ · v, and the subspace inclusion Q_{4} : ker(ϕ ∘ Q_{1}) → V_{1}. The projections are P_{3}^{α} = α · ϕ ∘ Q_{1} : V_{1} → 𝕂, P_{4} = Id_{V_{1}} − Q_{3}^{β} ∘ P_{3}^{α} : V_{1} → ker(ϕ ∘ Q_{1}). Theorem 1.75 applies to get the claimed direct sum. In particular, the first inclusion is Q_{1} ∘ Q_{3}^{β} : 𝕂 → V, the second is a subspace inclusion Q_{1} ∘ Q_{4} : ker(ϕ ∘ Q_{1}) → V, and the third is the subspace inclusion not depending on ϕ, Q_{2} : V_{2} ↪ V. The three projections are α·ϕ∘Q_{1}∘P_{1} : V → 𝕂, (Id_{V_{1}} −Q_{3}^{β} ∘(α·ϕ∘Q_{1}))∘P_{1} : V → ker(ϕ∘Q_{1}), and P_{2} : V ↠ V_{2}.

For K and ϕ as in Theorem 1.119, any element u ∈ V can be written in the following way as a sum of terms that do not depend on α or β:

u = Id_{V} (u) (1.11)

= (Q_{1} ∘ Q_{3}^{β} ∘ (α · ϕ ∘ Q_{1} ∘ P_{1})
+Q_{1} ∘ Q_{4} ∘ (Id_{V_{1}} − Q_{3}^{β} ∘ (α · ϕ ∘ Q_{1})) ∘ P_{1} + Q_{2} ∘ P_{2})(u)
= \frac{ϕ(u)}{2ϕ(w)}(w + K(w)) + (\frac{1}{2}(u + K(u)) − \frac{ϕ(u)}{2ϕ(w)}(w + K(w))) + \frac{1}{2}(u − K(u)).

The second and third terms are in the kernel of ϕ; the third term is the projection of u onto the −1 eigenspace of K, not depending on ϕ or w. The first two terms are both in the +1 eigenspace of K, and they both depend on ϕ and on v = \frac{1}{2} · (w + K(w)).

Lemma 1.120. Given \frac{1}{2} ∈ 𝕂 and two involutions K : V → V and K′ : U → U, which produce direct sums V_{1} ⊕ V_{2}, U_{1} ⊕ U_{2} as in Lemma 1.114, a map H : U → V respects the direct sums U_{1} ⊕ U_{2} → V_{1} ⊕ V_{2} if and only if K ∘ H = H ∘ K′.

Lemma 1.121. Given \frac{1}{2} ∈ 𝕂 and two involutions K : V → V and K′ : U → U, which produce direct sums V_{1} ⊕ V_{2}, U_{1} ⊕ U_{2} as in Lemma 1.114, a map H : U → V respects the direct sums U_{1} ⊕ U_{2} → V_{2} ⊕ V_{1} if and only if K ∘ H = −H ∘ K′.

Proof. Note the order of the spaces V_{2} ⊕ V_{1} is different from that appearing in Lemma 1.120, so the notation refers to the identities Q_{1} ∘ P_{1} ∘ H = H ∘ Q′_{2} ∘ P′_{2} and Q_{2} ∘ P_{2} ∘ H = H ∘ Q′_{1} ∘ P′_{1}. The claims can be checked directly, but also follow from applying Lemma 1.120 to the involutions K and −K′.

Lemma 1.122. Given V and a pair of involutions on V, K_{1} and K_{2}, if \frac{1}{2} ∈ 𝕂, then the following are equivalent.

(1) The involutions commute, i.e., K_{1} ∘ K_{2} = K_{2} ∘ K_{1}.

(2) The composite K_{1} ∘ K_{2} is an involution.

(3) K_{2} respects the direct sum V = V_{1} ⊕ V_{2} produced by K_{1}.

(4) K_{1} respects the direct sum V = V_{3} ⊕ V_{4} produced by K_{2}.

Proof. The equivalence (1) ⟺ (2) is elementary and does not require \frac{1}{2} ∈ 𝕂. The direct sums in (3), (4) are as in Lemma 1.114. The equivalences (1) ⟺ (3) and (1) ⟺ (4) are special cases of Lemma 1.120.

In statement (3) of Lemma 1.122, K_{2} induces an involution on both V_{1} and V_{2} as in Definition 1.84 and Lemma 1.85, and similarly for K_{1} in statement (4).

Given \frac{1}{2} ∈ 𝕂, and V with commuting involutions K_{1}, K_{2} as in Lemma 1.122 and Lemma 1.123, and corresponding direct sums V = V_{1} ⊕ V_{2}, V = V_{3} ⊕ V_{4}, respectively, as in Lemma 1.122, let V = V_{5} ⊕ V_{6} be the direct sum produced by the involution K_{1} ∘ K_{2}.

Lemma 1.123. Given V, subspaces V_{1}, …, V_{5} as above, and commuting involutions K_{1}, K_{2} ∈ End(V), for v ∈ V, any pair of two of the following three statements implies the remaining one:

(1) v = K_{1}(v) ∈ V_{1}.

(2) v = K_{2}(v) ∈ V_{3}.

(3) v = (K_{1} ∘ K_{2})(v) ∈ V_{5}.

It follows from Lemma 1.123 that these subspaces of V are equal:

(1.12)  V_{1} ∩ V_{3} = V_{1} ∩ V_{5} = V_{3} ∩ V_{5} = V_{1} ∩ V_{3} ∩ V_{5}.

Let (P_{5}, P_{6}), (Q_{5}, Q_{6}) denote the projections and inclusions for the above direct sum V = V_{5} ⊕V_{6} produced by K_{1} ∘K_{2}. Since K_{1} ∘Q_{5} = K_{2} ∘Q_{5}, the maps induced by K_{1} and K_{2} are equal:

(1.13)  P_{5} ∘ K_{1} ∘ Q_{5} = P_{5} ∘ K_{2} ∘ Q_{5} : V_{5} → V_{5};

this map is a canonical involution on V_{5}, producing a direct sum V_{5} = V′_{5} ⊕ V′′_{5} with projection P′_{5} : V_{5} ↠ V′_{5} from (1.9). Similarly, there is an involution induced by K_{1} or K_{1} ∘ K_{2} on V_{3}, producing V_{3} = V′_{3} ⊕ V′′_{3}, and there is another involution induced by K_{2} or K_{1} ∘ K_{2} on V_{1}, producing V_{1} = V′_{1} ⊕ V′′_{1}. On the set V_{6}, the induced involutions are opposite:

(1.14)  P_{6} ∘ K_{1} ∘ Q_{6} = −P_{6} ∘ K_{2} ∘ Q_{6} : V_{6} → V_{6},

if one produces a direct sum V_{6} = V′_{6} ⊕ V′′_{6}, the other produces V_{6} = V′′_{6} ⊕ V′_{6}. Similarly, there are opposite induced involutions on V_{2} and V_{4}.

Theorem 1.124. Given \frac{1}{2} ∈ 𝕂, and commuting involutions on V with the above notation,

V′_{5} = V′_{3} = V′_{1} = V_{1} ∩ V_{3} ∩ V_{5}.

The composite projections are all equal:

P′_{5} ∘ P_{5} = P′_{3} ∘ P_{3} = P′_{1} ∘ P_{1} : V ↠ V_{1} ∩ V_{3} ∩ V_{5}.

Also, V′′_{5} = V_{2} ∩ V_{4}, V′′_{3} = V_{2} ∩ V_{6}, and V′′_{1} = V_{4} ∩ V_{6}.

Proof. V′_{5} is the set of fixed points v ∈ V_{5} of the involution P_{5} ∘ K_{1} ∘ Q_{5}. Denote the operators from Lemma 1.114 P′_{5}, Q′_{5}, so

Q′_{5} ∘ P′_{5} = \frac{1}{2} · (Id_{V_{5}} + P_{5} ∘ K_{1} ∘ Q_{5}).

To establish the first claim, it is enough to show V′_{5} = V_{1}∩V_{3}; the claims V′_{3} = V_{1}∩V_{5} and V′_{1} = V_{3} ∩ V_{5} are similar, and then (1.12) applies. To show V′_{5} ⊆ V_{1}, use the fact that K_{1} commutes with Q_{5} ∘ P_{5} = \frac{1}{2} · (Id_{V} + K_{1} ∘ K_{2}); if v ∈ V′_{5} ⊆ V_{5}, then

v = Q_{5}(v) = (P_{5} ∘ K_{1} ∘ Q_{5})(v) = Q_{5}((P_{5} ∘ K_{1} ∘ Q_{5})(v)) = K_{1}(Q_{5}(v)),

so v ∈ V_{1}. Showing V′_{5} ⊆ V_{3} is similar, so V′_{5} ⊆ V_{1} ∩ V_{3}.

Another argument would be to consider the subspace V′_{5} as the image of Q_{5} ∘ Q′_{5} ∘ P′_{5} ∘ P_{5} in V. Then

Q_{5} ∘ Q′_{5} ∘ P′_{5} ∘ P_{5} = Q_{5} ∘ \frac{1}{2} · (Id_{V_{5}} + P_{5} ∘ K_{1} ∘ Q_{5}) ∘ P_{5}
= \frac{1}{2} · Q_{5} ∘ P_{5} + \frac{1}{2} · K_{1} ∘ Q_{5} ∘ P_{5}
(1.15)  = Q_{1} ∘ P_{1} ∘ Q_{5} ∘ P_{5},

which shows V′_{5} is contained in V_{1}, the image of Q_{1} in V.

Conversely, if v ∈ V_{1} ∩ V_{3}, then v = K_{1}(v) = Q_{5}(v) ∈ V_{5} (Lemma 1.123) and (P_{5} ∘ K_{1} ∘ Q_{5})(v) = (P_{5} ∘ Q_{5})(v) = v ∈ V′_{5}.

The equality of the composites of projections follows from using the commutativity of the involutions to get Q_{1} ∘ P_{1} ∘ Q_{5} ∘ P_{5} = Q_{5} ∘ P_{5} ∘ Q_{1} ∘ P_{1}, and then (1.15) implies Q_{5} ∘ Q′_{5} ∘ P′_{5} ∘ P_{5} = Q_{1} ∘ Q′_{1} ∘ P′_{1} ∘ P_{1}.

The last claim of the Theorem follows from similar calculations. However, the three subspaces are in general not equal to each other.

The projection P_{5} : V ↠ V_{5} satisfies P_{5} ∘ K_{1} = (P_{5} ∘ K_{1} ∘ Q_{5}) ∘ P_{5}, so Lemma 1.120 applies: P_{5} respects the direct sums V_{1} ⊕V_{2} → V′_{5} ⊕V′′_{5} and the map V_{1} → V′_{5} induced by P_{5} is P′_{5} ∘ P_{5} ∘ Q_{1}. By Theorem 1.124,

(1.16)  P′_{5} ∘ P_{5} ∘ Q_{1} = P′_{1} ∘ P_{1} ∘ Q_{1} = P′_{1} : V_{1} ↠ V′_{1} = V′_{5}.

This gives an alternate construction of P′_{1} as a map induced by P_{5}, or similarly, any P′_{i} is equal to a map induced by P_{I} for any distinct i = 1, 3, 5, I = 1, 3, 5.

Theorem 1.125. Given \frac{1}{2} ∈ 𝕂, suppose K_{V}^{1} , K_{V}^{2} are commuting involutions on V as in Theorem 1.124. Similarly, let K_{U}^{1}, K_{U}^{2} be commuting involutions on U, with corresponding notation for the direct sums: U = U_{1} ⊕ U_{2}, U = U_{3} ⊕ U_{4}, etc. If a map H : U → V satisfies H ∘K_{U}^{1} = K_{V}^{1} ∘H and H ∘K_{U}^{2} = K_{V}^{2} ∘H, then H respects the corresponding direct sums U_{1} ⊕ U_{2} → V_{1} ⊕ V_{2} and U_{3} ⊕ U_{4} → V_{3} ⊕ V_{4}. Further, the induced map P_{V}^{1} ∘H ∘Q_{U}^{1} : U_{1} → V_{1} respects the direct sums U′_{1} ⊕U′′_{1} → V′_{1} ⊕V′′_{1} and similarly for the maps U_{3} → V_{3}, U_{5} → V_{5} induced by H. The induced map U′_{1} → V′_{1} is equal to the map U′_{3} → V′_{3} induced by P_{V}^{3} ∘ H ∘ Q_{U}^{3} : U_{3} → V_{3}.

Proof. The fact that H respects each pair of direct sums is Lemma 1.120. The subspace U_{1} has a canonical involution P_{U}^{1} ∘ K_{U}^{2} ∘ Q_{U}^{1}, and since K_{U}^{1}, K_{U}^{2} commute, K_{U}^{2} also commutes with Q_{U}^{1} ∘ P_{U}^{1} = \frac{1}{2} · (Id_{U} + K_{U}^{1}). The map induced by H, P_{V}^{1} ∘ H ∘ Q_{U}^{1} : U_{1} → V_{1}, satisfies:

(P_{V}^{1} ∘ H ∘ Q_{U}^{1}) ∘ (P_{U}^{1} ∘ K_{U}^{2} ∘ Q_{U}^{1}) = P_{V}^{1} ∘ H ∘ K_{U}^{2} ∘ Q_{U}^{1} ∘ P_{U}^{1} ∘ Q_{U}^{1}
= P_{V}^{1} ∘ K_{V}^{2} ∘ H ∘ Q_{U}^{1}
= (P_{V}^{1} ∘ K_{V}^{2} ∘ Q_{V}^{1} ) ∘ (P_{V}^{1} ∘ H ∘ Q_{U}^{1}).

It follows from Lemma 1.120 again that P_{V}^{1} ∘ H ∘ Q_{U}^{1} respects the direct sums as claimed. The induced map is P_{V}^{1′} ∘ (P_{V}^{1} ∘ H ∘ Q_{U}^{1}) ∘ Q_{U}^{1′} : U′_{1} → V′_{1}.

The last claim of the Theorem is that this induced map is equal to the map P_{V}^{3′}∘(P_{V}^{3} ∘H ∘Q_{U}^{3})∘Q_{U}^{3′} : U′_{3} → V′_{3}. The claim follows from the idea that the induced maps are restrictions of H to the same subspace U′_{1} = U′_{3} = U_{1} ∩ U_{3}, by Theorem 1.124. More specifically, the subspace inclusions are equal: Q_{U}^{3} ∘ Q_{U}^{3′} = Q_{U}^{1} ∘ Q_{U}^{1′} : U_{1} ∩ U_{3} ↪ U, and the composites of projections are equal: P_{V}^{1′} ∘ P_{V}^{1} = P_{V}^{3′} ∘ P_{V}^{3} .

Example 1.126. Given \frac{1}{2} ∈ 𝕂, suppose K_{1} and K_{2} are commuting involutions on V as in Theorem 1.124, and suppose H is another involution on V so that K_{1} ∘ H = H ∘ K_{2}. The three involutions K_{1}, K_{2}, K_{1} ∘ K_{2} produce direct sums V = V_{1} ⊕ V_{2}, V_{3} ⊕ V_{4}, and V_{5} ⊕ V_{6}. Similarly, because K_{1} ∘ K_{2} commutes with H, the three involutions K_{1} ∘ K_{2}, H, and K_{1} ∘ K_{2} ∘ H produce corresponding direct sums V = V_{5} ⊕V_{6}, V_{7} ⊕V_{8}, and V_{9} ⊕V_{10}. As in (1.13), there are induced involutions on V_{5}, P_{5} ∘ K_{1} ∘ Q_{5} = P_{5} ∘ K_{2} ∘ Q_{5} and P_{5} ∘ H ∘ Q_{5} = P_{5} ∘ K_{1} ∘ K_{2} ∘ H ∘ Q_{5}. These two involutions commute: for v = Q_{5}(v) = (K_{1} ∘ K_{2})(v) ∈ V_{5}, v satisfies K_{1}(v) = K_{2}(v) and

(P_{5} ∘ K_{1} ∘ Q_{5} ∘ P_{5} ∘ H ∘ Q_{5})(v) = (P_{5} ∘ K_{1} ∘ H)(v) = (P_{5} ∘ H ∘ K_{2})(v),
(P_{5} ∘ H ∘ Q_{5} ∘ P_{5} ∘ K_{1} ∘ Q_{5})(v) = (P_{5} ∘ H ∘ K_{1})(v).

By Lemma 1.122, their product

(P_{5} ∘ K_{1} ∘ Q_{5}) ∘ (P_{5} ∘ H ∘ Q_{5}) = P_{5} ∘ K_{1} ∘ H ∘ Q_{5} = P_{5} ∘ K_{2} ∘ H ∘ Q_{5}

is also an involution on V_{5} (although in general, K_{1} ∘ H and K_{2} ∘ H need not be involutions). So, Theorem 1.124 applies to these three commuting involutions on V_{5}, with P_{5} ∘K_{1} ∘Q_{5} producing a direct sum V_{5} = V′_{5} ⊕V′′_{5}, where V′_{5} = V_{1} ∩V_{3} ∩V_{5} is the fixed point subspace of P_{5} ∘ K_{1} ∘ Q_{5}. Similarly, V_{5} ∩ V_{7} ∩ V_{9} is the fixed point subspace of P_{5} ∘ H ∘ Q_{5}, and denoting by V_{11} the fixed point subspace of P_{5} ∘ K_{1} ∘ H ∘ Q_{5}, the three fixed point subspaces have the following intersection:

(V_{1} ∩ V_{3} ∩ V_{5}) ∩ (V_{5} ∩ V_{7} ∩ V_{9}) ∩ V_{11} = (V_{1} ∩ V_{3}) ∩ (V_{5} ∩ V_{7})
= (V_{1} ∩ V_{3} ∩ V_{5}) ∩ V_{7} = V_{1} ∩ V_{3} ∩ V_{7}
= {v ∈ V : v = K_{1}(v) = K_{2}(v) = H(v)}.

The projections from Theorem 1.124 appear in the following commutative diagram.

\xymatrix{ &  & V \ar@{->>}[dll] \ar@{->>}[dl] \ar@{->>}[d] \ar@{->>}[dr] \ar@{->>}[drr] \\ V_{1} \ar@{->>}[dr] & V_{3} \ar@{->>}[d] & V_{5} \ar@{->>}[dl] \ar@{->>}[d] \ar@{->>}[dr] & V_{7} \ar@{->>}[d] & V_{9} \ar@{->>}[dl] \\  & V_{1} ∩ V_{3} ∩ V_{5} \ar@{->>}[dr] & V_{11} \ar@{->>}[d] & V_{5} ∩ V_{7} ∩ V_{9} \ar@{->>}[dl] \\  &  & V_{1} ∩ V_{3} ∩ V_{7}}

Example 1.127. The construction in Example 1.126 also works under the hypothesis that H commutes with both K_{1} and K_{2} (instead of K_{1} ∘ H = H ∘ K_{2}).

Exercise 1.128. For involutions on V as in Example 1.126 satisfying K_{1}∘K_{2} = K_{2} ∘ K_{1} and K_{1} ∘ H = H ∘ K_{2}, the set

{Id_{V} , K_{1}, K_{2}, K_{1} ∘ K_{2}, H, K_{1} ∘ H, K_{2} ∘ H, K_{1} ∘ K_{2} ∘ H}

is the image of a representation D_{4} ⇝ End(V), where D_{4} is the eight-element dihedral group.

Remark 1.129. An example of a vector space over 𝕂 = ℚ admitting three involutions satisfying the relations of Example 1.126 is ℚ(\sqrt{−1}, \sqrt[4]{2}), as considered by [Cox] (Example 7.3.4), along with diagrams analogous to the above diagram.

Theorem 1.130. Given V, the following statements (1) to (7) are equivalent, and any implies (8). Further, if \frac{1}{2} ∈ 𝕂, then all eight statements are equivalent.

(1) V admits a direct sum of the form V = U ⊕ U.

(2) V = U_{1} ⊕ U_{2} and there exist invertible maps A_{1} : U_{3} → U_{1} and A_{2} : U_{3} → U_{2}.

(3) V = U′ ⊕ U′′ and there exists an invertible map A : U′′ → U′.

(4) V = U′ ⊕ U′′ and there exists an involution K ∈ End(V) that respects the direct sums U′ ⊕ U′′ → U′′ ⊕ U′.

(5) V admits an idempotent P ∈ End(V) and an involution K ∈ End(V) such that P ∘ K = K ∘ (Id_{V} − P).

(6) V = U′ ⊕ U′′ and there exists an invertible H ∈ End(V) that respects the direct sums U′ ⊕ U′′ → U′′ ⊕ U′.

(7) V admits an idempotent P ∈ End(V) and an invertible map H ∈ End(V) such that P ∘ H = H ∘ (Id_{V} − P).

(8) V admits anticommuting involutions K_{1}, K_{2} (i.e., K_{1} ∘ K_{2} = −K_{2} ∘ K_{1}).

Proof. The implication (1) ⟹ (2) is canonical: let U_{1} = U_{2} = U_{3} = U, and A_{1} = A_{2} = Id_{U}.

The implication (2) ⟹ (1) is canonical. Given V = U_{1} ⊕ U_{2} with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), Example 1.93 applies. Let U = U_{3}, to get V = U ⊕ U with projections (A_{1}^{−1} ∘ P_{1}, A_{2}^{−1} ∘ P_{2}) and inclusions (Q_{1} ∘ A_{1}, Q_{2} ∘ A_{2}). The direct sums are equivalent.

The implication (1) ⟹ (3) is canonical: let U′ = U′′ = U and A = Id_{U}.

The implication (2) ⟹ (3) is canonical: let U′ = U_{1}, U′′ = U_{2} and A = A_{1} ∘ A_{2}^{−1}.

For (3) ⟹ (1), there are two choices. Given V = U′ ⊕ U′′ with projections (P′, P′′) and inclusions (Q′, Q′′), one choice is to let U = U′. Then V = U ⊕ U with projections (P_{1}, A ∘ P_{2}) and inclusions (Q_{1}, Q_{2} ∘ A^{−1}). The other choice is to let U = U′′, with projections (A^{−1} ∘ P_{1}, P_{2}) and inclusions (Q_{1} ∘ A, Q_{2}). As in Example 1.93, either of the two constructions gives a direct sum equivalent to V = U′ ⊕ U′′, so they are equivalent to each other.

For (3) ⟹ (2), there are two choices. One choice is to let U_{3} = U′, A_{1} = Id_{U′}, A_{2} = A^{−1}. Applying the canonical (2) ⟹ (1) construction then gives projections (P_{1}, A ∘ P_{2}) as in the first choice of the previous implication. The second choice is to let U_{3} = U′′, A_{1} = A, A_{2} = Id_{U′′}, which similarly corresponds to the second choice in the previous implication.

The implication (3) ⟹ (4) is canonical. Given A : U′′ → U′, let

(1.17)  K = Q′′ ∘ A^{−1} ∘ P′ + Q′ ∘ A ∘ P′′.

It is straightforward to check that K is an involution, and Q′ ∘P′ ∘K = K ∘Q′′ ∘P′′, so K respects the direct sums as in Example 1.90.

The implication (4) ⟹ (3) is canonical. Given K, let A = P′ ∘ K ∘ Q′′ : U′′ → U′, which by Lemma 1.85 has inverse A^{−1} = P′′ ∘ K ∘ Q′.

The implication (4) ⟹ (6) is canonical: let H = K.

For (6) ⟹ (3), there are two choices. One choice is to let A = P′ ∘ H ∘ Q′′ : U′′ → U′, so A^{−1} = P′′ ∘ H^{−1} ∘ Q′. The canonical involution (1.17) from the implication (3) ⟹ (4) is then K = Q′′ ∘ P′′ ∘ H^{−1} ∘ Q′ ∘ P′ + Q′ ∘ P′ ∘ H ∘ Q′′ ∘ P′′. The second choice is to let A = P′ ∘ H^{−1} ∘ Q′′. This similarly leads to an involution Q′′ ∘ P′′ ∘ H ∘ Q′ ∘ P′ + Q′ ∘ P′ ∘ H^{−1} ∘ Q′′ ∘ P′′, which, unless H is an involution, may be different from the involution from the first choice.

For (4) ⟹ (5), and for (6) ⟹ (7), there are two choices: P = Q′ ∘ P′, or P = Q′′ ∘ P′′. This choice between two idempotents was already mentioned in Example 1.110.

Conversely, for (5) ⟹ (4) (and similarly for (7) ⟹ (6)), there are two choices. For U′ = ker(P) and U′′ = P (V), as in Example 1.109, there are two ways to form a direct sum: V = U′ ⊕ U′′ or V = U′′ ⊕ U′. The map K (similarly H) respects the direct sums as in Lemma 1.111.

For (4) ⟹ (8), which does not require \frac{1}{2} ∈ 𝕂, there are two choices (assuming the ordering of the pair K_{1}, K_{2} does not matter). Given K, let K_{1} = K. One choice is to let K_{2} = Q′ ∘ P′ − Q′′ ∘ P′′, as in Example 1.116. It follows from K ∘ Q′ ∘ P′ = Q′′ ∘ P′′ ∘ K that K_{1} ∘ K_{2} = −K_{2} ∘ K_{1}. The second choice is to let K_{2} = −Q′ ∘ P′ + Q′′ ∘ P′′.

Similarly for (5) ⟹ (8), there are two choices. Given K, let K_{1} = K. One choice is to let K_{2} = 2 · P − Id_{V}, as in Lemma 1.117. The second choice is to let K_{2} = Id_{V} − 2 · P.

For (8) ⟹ (4) using \frac{1}{2} ∈ 𝕂, the involution K_{1} produces a direct sum V = V_{1} ⊕ V_{2} as in Lemma 1.114, with projections P_{1} = \frac{1}{2}(Id_{V} + K_{1}), P_{2} = \frac{1}{2}(Id_{V} − K_{1}) (the order of the direct sum could be chosen the other way, V_{2} ⊕ V_{1}). By Lemma 1.121, K = K_{2} satisfies (4). In this case, the invertible map from (3) is, by Lemma 1.85, the composite

(1.18)  A = P_{1} ∘ K_{2} ∘ Q_{2} : V_{2} → V_{1},

with inverse P_{2} ∘ K_{2} ∘ Q_{1} : V_{1} → V_{2}. Another choice for (8) ⟹ (4) is to use K_{2} to produce a different direct sum, and then let K = K_{1}.

Theorem 1.131. Given \frac{1}{2} ∈ 𝕂 and two involutions K, K′ ∈ End(V), which produce direct sums V = V_{1} ⊕ V_{2}, V = V′_{1} ⊕ V′_{2} as in Lemma 1.114, if K and K′ anticommute, then for i = 1, 2, I = 1, 2, and β ∈ 𝕂, β ≠ 0, the map

β · P′_{I} ∘ Q_{i} : V_{i} → V′_{I}

is invertible.

Proof. Consider P′_{I} ∘ Q_{i} : V_{i} → V′_{I} and P_{i} ∘ Q′_{I} : V′_{I} → V_{i}. Then

P′_{I} ∘ Q_{i} ∘ P_{i} ∘ Q′_{I} = P′_{I} ∘ \frac{1}{2} · (Id_{V} ± K) ∘ Q′_{I}.

Since K respects the direct sums V′_{1} ⊕V′_{2} → V′_{2} ⊕V′_{1} by Lemma 1.121, P′_{I} ∘K ∘Q′_{I} = 0_{End(V′_{I})} by Lemma 1.83. In the other order,

P_{i} ∘ Q′_{I} ∘ P′_{I} ∘ Q_{i} = P_{i} ∘ \frac{1}{2} · (Id_{V} ± K′) ∘ Q_{i},

and similarly, P_{i} ∘ K′ ∘ Q_{i} = 0_{End(V_{i})}.

Since P′_{I} ∘ Q′_{I} = Id_{V′_{I}} and P_{i} ∘ Q_{i} = Id_{V_{i}}, the conclusion is that for any scalar β ∈ 𝕂, β ≠ 0, the map β · P′_{I} ∘ Q_{i} has inverse \frac{2}{β} · P_{i} ∘ Q′_{I}.

Lemma 1.132. Given involutions K_{1}, K_{2}, K_{3} ∈ End(V), any pair of two of the following three statements implies the remaining one.

(1) K_{3} commutes with K_{1} ∘ K_{2}.

(2) K_{3} anticommutes with K_{1}.
(3) K_{3} anticommutes with K_{2}.

Exercise 1.133. If K_{1}, K_{2}, K_{3} are involutions such that K_{1} and K_{2} commute and K_{3} satisfies the three conditions from Lemma 1.132, then the set

{±Id_{V} , ±K_{1}, ±K_{2}, ±K_{3}, ±K_{2} ∘ K_{3}, ±K_{1} ∘ K_{3}, ±K_{1} ∘ K_{2}, ±K_{1} ∘ K_{2} ∘ K_{3}}

is the image of a representation D_{4} × ℤ_{2} ⇝ End(V), where D_{4} is the eight-element dihedral group and ℤ_{2} is the two-element group.

For \frac{1}{2} ∈ 𝕂 and commuting involutions K_{1}, K_{2}, recall the direct sums V = V_{1} ⊕ V_{2}, V = V_{3} ⊕ V_{4}, V = V_{5} ⊕ V_{6} from Lemma 1.123 produced by K_{1}, K_{2}, K_{1} ∘ K_{2}. Further, suppose K_{3} is another involution satisfying the three conditions from Lemma 1.132, and let V = V_{7} ⊕ V_{8} and V = V_{9} ⊕ V_{10} be the direct sums produced by the involutions K_{3} and K_{1} ∘ K_{2} ∘ K_{3}. Theorem 1.124 applies to V_{5} twice: first, to the pair K_{1}, K_{2} to get the canonical involution P_{5} ∘ K_{1} ∘ Q_{5} from (1.13) producing V_{5} = V′_{5} ⊕ V′′_{5} with V′_{5} = V_{1} ∩ V_{3} ∩ V_{5}, and second, to the other pair K_{1} ∘ K_{2}, K_{3} to get another involution P_{5} ∘ K_{3} ∘ Q_{5} = P_{5} ∘ K_{1} ∘ K_{2} ∘ K_{3} ∘ Q_{5} as in (1.13), producing another direct sum V_{5} = V′′′_{5} ⊕ V′′′′_{5}, with V′′′_{5} = V_{5} ∩ V_{7} ∩ V_{9}.

Corollary 1.134. Given \frac{1}{2} ∈ 𝕂, 0 ≠ β ∈ 𝕂, commuting involutions K_{1}, K_{2}, and an involution K_{3} as in Lemma 1.132, the map

β · P′′′_{5} ∘ Q′_{5} : V′_{5} → V′′′_{5}

is invertible.

Proof. Q′_{5} is as in the Proof of Theorem 1.124. The projection P′′′_{5} : V_{5} ↠ V′′′_{5} is from the direct sum V_{5} = V′′′_{5} ⊕ V′′′′_{5} produced by P_{5} ∘ K_{3} ∘ Q_{5}. The involutions P_{5} ∘ K_{1} ∘ Q_{5}, P_{5} ∘ K_{3} ∘ Q_{5} ∈ End(V_{5}) anticommute, and Theorem 1.131 applies.

Using a step analogous to (1.15), the output of the above invertible map, for input v ∈ V′_{5}, can be written as:

β · P′′′_{5} ∘ Q′_{5} : v ↦ β · (P′′′_{5} ∘ Q′_{5})(v)
(1.19)  = Q_{5}(Q′′′_{5}(β · (P′′′_{5} ∘ (P_{5} ∘ Q_{5}) ∘ Q′_{5})(v)))
= β · ((Q_{5} ∘ Q′′′_{5} ∘ P′′′_{5} ∘ P_{5}) ∘ Q_{5} ∘ Q′_{5})(v)
= β · ((Q_{7} ∘ P_{7} ∘ Q_{5} ∘ P_{5}) ∘ Q_{5} ∘ Q′_{5})(v)
= β · (Q_{7} ∘ P_{7} ∘ Q_{5} ∘ Q′_{5})(v)
= \frac{β}{2} · (v + K_{3}(v)).

Corollary 1.134 could be re-stated as constructing an invertible map between these subspaces of V_{5} = {v ∈ V : v = (K_{1} ∘ K_{2})(v)}:

{v = K_{1}(v) = K_{2}(v)} → {v = K_{3}(v) = (K_{1} ∘ K_{2} ∘ K_{3})(v)}.

Two more subspaces of V_{5}, from Theorem 1.124, are:

V′′_{5} = {v ∈ V : v = −K_{1}(v) = −K_{2}(v)} = V_{2} ∩ V_{4},
V′′′′_{5} = {v ∈ V : v = −K_{3}(v) = −(K_{1} ∘ K_{2} ∘ K_{3})(v)} = V_{8} ∩ V_{10},

and Theorem 1.131 also gives a construction of invertible maps: V′_{5} → V′′′′_{5}, V′′_{5} → V′′′_{5}, and V′′_{5} → V′′′′_{5}.

Example 1.135. Given any spaces V and W, and an involution K on V, the map [Id_{W} ⊗ K] is an involution on W ⊗ V. If \frac{1}{2} ∈ 𝕂, then the direct sum produced by [Id_{W} ⊗ K] has projections \frac{1}{2} · (Id_{W ⊗V} ± [Id_{W} ⊗ K]). For the direct sum V = V_{1}⊕V_{2} as in Lemma 1.114 with inclusions Q_{1}, Q_{2}, there is also a direct sum W ⊗V = W ⊗V_{1}⊕W ⊗V_{2} as in Example 1.77, with projections [Id_{W} ⊗ \frac{1}{2} ·(Id_{V} ±K)] and inclusions [Id_{W} ⊗ Q_{i}]. The two constructions lead to the same formula for the projection operators, so the projections are canonical and K produces a direct sum W ⊗ V = W ⊗ V_{1} ⊕ W ⊗ V_{2}. The space W ⊗ V_{1} is a subspace of W ⊗ V, equal to the fixed point set of [Id_{W} ⊗ K], with inclusion map [Id_{W} ⊗ Q_{1}], and similarly W ⊗ V_{2} is the fixed point subspace of −[Id_{W} ⊗ K]. The space V ⊗ W admits an analogous involution and direct sum.

Example 1.136. Given any spaces U, W with involutions K_{U} on U and K_{W} on W, the involutions [Id_{U} ⊗ K_{W}] and [K_{U} ⊗ Id_{W}] on U ⊗ W commute, so Lemma 1.123 applies, and if \frac{1}{2} ∈ 𝕂, then Lemma 1.122 and Theorem 1.124 apply. For the direct sums U = U_{1} ⊕ U_{2} and W = W_{1} ⊕ W_{2} produced as in Lemma 1.114, [K_{U} ⊗ Id_{W}] respects the direct sum U ⊗ W_{1} ⊕ U ⊗ W_{2} from Example 1.135; the induced involution on U ⊗ W_{1} is exactly [K_{U} ⊗ Id_{W_{1}}], so U ⊗ W_{1} admits a direct sum U_{1} ⊗ W_{1} ⊕ U_{2} ⊗ W_{1}. Similarly, [Id_{U} ⊗ K_{W}] induces an involution on U_{1} ⊗ W and a direct sum U_{1} ⊗ W_{1} ⊕ U_{1} ⊕ W_{2}. The subspace U_{1} ⊗ W_{1} appears in two different ways, but there is no conflict in naming it: by Theorem 1.124, U_{1} ⊗ W_{1} = (U ⊗ W_{1}) ∩ (U_{1} ⊗ W).

Example 1.137. Given any spaces V and W, and an involution K on V, the map Hom(Id_{W} , K) is an involution on Hom(W, V). If \frac{1}{2} ∈ 𝕂, then the direct sum produced by Hom(Id_{W} , K) has projections

\frac{1}{2} · (Id_{Hom(W,V)} ± Hom(Id_{W} , K)) : A ↦ \frac{1}{2} · (A ± K ∘ A).

For the direct sum V = V_{1} ⊕ V_{2} as in Lemma 1.114, there is also a direct sum Hom(W, V) = Hom(W, V_{1}) ⊕ Hom(W, V_{2}) as in Example 1.78, with projections

Hom(Id_{W} , P_{i}) : Hom(W, V) → Hom(W, V_{i}) : A ↦ P_{i} ∘ A = \frac{1}{2} · (Id_{V} ± K) ∘ A.

The two constructions lead to the same formula for the projection operators. The only difference is in the target space: the fixed point set of Hom(Id_{W} , K) is the set of maps A : W → V such that A = K ∘ A, while the image of the projection Hom(Id_{W} , P_{1}) is a set of maps with domain W and target V_{1} = {v ∈ V : v = K(v)}, which is a subspace of V. It will not cause any problems to consider Hom(W, V_{i}) as a subspace of Hom(W, V); more precisely, in the case where V = V_{1} ⊕ V_{2} is a direct sum produced by an involution, the operator Hom(Id_{W} , Q_{i}) from Example 1.78 can be regarded as a subspace inclusion as in Lemma 1.114, so A and Q_{i} ∘ A are identified. Then the above two direct sum constructions have the same projection and inclusion operators, so the projections are canonical and K produces a direct sum Hom(W, V) = Hom(W, V_{1}) ⊕ Hom(W, V_{2}).

Example 1.138. Given any spaces V and W, and an involution K on V, the map Hom(K, Id_{W}) is an involution on Hom(V, W). If \frac{1}{2} ∈ 𝕂, then the direct sum produced by Hom(K, Id_{W}) has projections

(1.20)  \frac{1}{2} · (Id_{Hom(V,W)} ± Hom(K, Id_{W})) : A ↦ \frac{1}{2} · (A ± A ∘ K).

For the direct sum V = V_{1} ⊕ V_{2} as in Lemma 1.114, there is also a direct sum Hom(V, W) = Hom(V_{1}, W) ⊕ Hom(V_{2}, W) as in Example 1.79, with projections

Hom(Q_{i}, Id_{W}) : Hom(V, W) → Hom(V_{i}, W) : A ↦ A ∘ Q_{i}.

Unlike Example 1.137, the two constructions lead to different formulas for the projection operators. The fixed point set of Hom(K, Id_{W}) is the set of maps A : V → W such that A = A ∘ K, while the image of the projection Hom(Q_{1}, Id_{W}) is a set of maps with domain V_{1} and target W — which does not look like a subspace of Hom(V, W). The conclusion is that the two direct sum constructions are different. However, they are equivalent, as in Definition 1.92. Checking statements (2) and (3) of Lemma 1.91,

Hom(P_{i}, Id_{W}) ∘ Hom(Q_{i}, Id_{W}) : A ↦ A ∘ Q_{i} ∘ P_{i} = A ∘ \frac{1}{2} · (Id_{V} ± K),

which is the same as (1.20).

Example 1.139. Given any spaces U, V, W, with involutions K_{U} on U, K_{V} on V, and Hom(K_{V} , Id_{W}) on Hom(V, W) as in Example 1.138, suppose \frac{1}{2} ∈ 𝕂 and H : U → Hom(V, W) satisfies Hom(K_{V} , Id_{W}) ∘ H = H ∘ K_{U}. Let U = U_{1} ⊕ U_{2} be the direct sum produced by K_{U}, and consider the direct sum on Hom(V, W) produced by Hom(K_{V} , Id_{W}) as in (1.20) from Example 1.138. Then, by Lemma 1.120, H respects the direct sums:

H : U_{1} ⊕ U_{2} → {A : V → W : A ∘ K_{V} = A} ⊕ {A : V → W : A ∘ K_{V} = −A}.

Let V = V_{1} ⊕ V_{2} be the direct sum produced by K_{V}; then by Lemma 1.94, H also respects the other, equivalent direct sum on Hom(V, W) from Example 1.138:

H : U_{1} ⊕ U_{2} → Hom(V_{1}, W) ⊕ Hom(V_{2}, W).

Example 1.140. Given \frac{1}{2} ∈ 𝕂, any spaces U and W, and involutions K_{U} ∈ End(U) and K_{W} ∈ End(W), let U = U_{1} ⊕ U_{2} be the direct sum produced by K_{U}, with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), and let W = W_{1} ⊕ W_{2} be the direct sum produced by K_{W}, with operators (P′_{1}, P′_{2}), (Q′_{1}, Q′_{2}). Then there are commuting involutions Hom(Id_{U}, K_{W}), Hom(K_{U}, Id_{W}), on Hom(U, W), and their composite Hom(K_{U}, K_{W}) is another involution. Theorem 1.124 applies, and we re-use its V_{1}, …, V_{6} notation for the produced direct sums. As in Example 1.137, there is a direct sum

Hom(U, W) = {A = K_{W} ∘ A} ⊕ {A = −K_{W} ∘ A}
= Hom(U, W_{1}) ⊕ Hom(U, W_{2})
= V_{1} ⊕ V_{2}.

As in Example 1.138, there are two different but equivalent direct sums, the first is

Hom(U, W) = {A = A ∘ K_{U}} ⊕ {A = −A ∘ K_{U}} = V_{3} ⊕ V_{4},

with projections and inclusions denoted (P_{3}, P_{4}), (Q_{3}, Q_{4}), and the second is

Hom(U, W) = Hom(U_{1}, W) ⊕ Hom(U_{2}, W),

with projections and inclusions

(Hom(Q_{1}, Id_{W}), Hom(Q_{2}, Id_{W})), (Hom(P_{1}, Id_{W}), Hom(P_{2}, Id_{W})).

From Lemma 1.95, there are canonical, invertible maps

P_{3} ∘ Hom(P_{1}, Id_{W}) : Hom(U_{1}, W) → V_{3},
Hom(Q_{1}, Id_{W}) ∘ Q_{3} : V_{3} → Hom(U_{1}, W).

There is also the direct sum produced by the composite involution,

Hom(U, W) = {A = K_{W} ∘ A ∘ K_{U}} ⊕ {A = −K_{W} ∘ A ∘ K_{U}}
= V_{5} ⊕ V_{6}.

It follows from Theorem 1.124 that V_{1}, V_{3}, and V_{5} admit canonical involutions and direct sums. For example, P_{3} ∘ Hom(Id_{U}, K_{W}) ∘ Q_{3} is the involution on V_{3}, and it produces the direct sum V_{3} = V′_{3} ⊕ V′′_{3}, where V′_{3} = V_{1} ∩ V_{3} ∩ V_{5}. The above invertible map Hom(Q_{1}, Id_{W}) ∘ Q_{3} : V_{3} → Hom(U_{1}, W) satisfies

Hom(Id_{U_{1}}, K_{W}) ∘ (Hom(Q_{1}, Id_{W}) ∘ Q_{3})
= Hom(Q_{1}, K_{W}) ∘ Q_{3}
= (Hom(Q_{1}, Id_{W}) ∘ Q_{3}) ∘ (P_{3} ∘ Hom(Id_{U}, K_{W}) ∘ Q_{3}) ,

so by Lemma 1.120, it respects the direct sums

V′_{3} ⊕ V′′_{3} → Hom(U_{1}, W_{1}) ⊕ Hom(U_{1}, W_{2}).

By Lemma 1.85, there is a canonical, invertible map from

V′_{3} = V_{1} ∩ V_{3} ∩ V_{5} = {A : U → W : A = K_{W} ∘ A = A ∘ K_{U} = K_{W} ∘ A ∘ K_{U}} to Hom(U_{1}, W_{1}), specifically, the map

A ↦ P′_{1} ∘ (Q_{3}(Q′_{3}(A))) ∘ Q_{1}
= P′_{1} ∘ A ∘ Q_{1} = \frac{1}{2} · (Id_{W} + K_{W}) ∘ A ∘ Q_{1} = A ∘ Q_{1}.

The inverse is defined for B ∈ Hom(U_{1}, W_{1}) by

B ↦ P′_{3}(P_{3}(Q′_{1} ∘ B ∘ P_{1})),

which, since Q′_{1} ∘ B ∘ P_{1} is an element of the subspace V_{1} ∩ V_{3}, simplifies to

Q′_{1} ∘ B ∘ P_{1} = B ∘ P_{1} = B ∘ \frac{1}{2} · (Id_{U} + K_{U}).

Example 1.141. For \frac{1}{2} ∈ 𝕂, and involutions K_{U} on U and K_{W} on W as in the previous Example, suppose K_{3} is an involution on Hom(U, W) that commutes with Hom(K_{U}, K_{W}) and anticommutes with either Hom(K_{U}, Id_{W}) or Hom(Id_{U}, K_{W}). Then Lemma 1.132 and Corollary 1.134 apply. Continuing with the V_{1}, …, V_{6} notation from Theorem 1.124 and Example 1.140, and also the V_{5} = V′′′_{5} ⊕ V′′′′_{5} and V_{7}, …, V_{10} notation from Corollary 1.134, the result of the Corollary is that for 0 ≠ β ∈ 𝕂, there is an invertible map:

β · P′′′_{5} ∘ Q′_{5} : V′_{5} → V′′′_{5}

which maps

{A ∈ Hom(U, W) : A = A ∘ K_{U} = K_{W} ∘ A}

to

{A ∈ Hom(U, W) : A = K_{W} ∘ A ∘ K_{U} = K_{3}(A)}.

There is also the canonical, invertible map from Example 1.140,

P′_{3} ∘ (P_{3} ∘ Hom(P_{1}, Id_{W})) ∘ Hom(Id_{U_{1}}, Q′_{1}),

which maps Hom(U_{1}, W_{1}) to V′_{3} = V′_{5}. The composite of these maps is an invertible map Hom(U_{1}, W_{1}) → V′′′_{5}:

(β · P′′′_{5} ∘ Q′_{5}) ∘ (P′_{3} ∘ (P_{3} ∘ Hom(P_{1}, Id_{W})) ∘ Hom(Id_{U_{1}}, Q′_{1})).

For B ∈ Hom(U_{1}, W_{1}), its output in V′′′_{5} ⊆ Hom(U, W) under the above map simplifies as follows, using the equality of subspace inclusions Q_{5} ∘ Q′_{5} = Q_{3} ∘ Q′_{3} and steps similar to (1.19):

B ↦ ((β · P′′′_{5} ∘ Q′_{5}) ∘ (P′_{3} ∘ P_{3} ∘ Hom(P_{1}, Q′_{1})))(B)
= Q_{5}(Q′′′_{5}(β · (P′′′_{5} ∘ (P_{5} ∘ Q_{5}) ∘ Q′_{5} ∘ P′_{3} ∘ P_{3} ∘ Hom(P_{1}, Q′_{1}))(B)))
= β · ((Q_{5} ∘ Q′′′_{5} ∘ P′′′_{5} ∘ P_{5}) ∘ (Q_{3} ∘ Q′_{3} ∘ P′_{3} ∘ P_{3}) ∘ Hom(P_{1}, Q′_{1}))(B)
= β · ((Q_{7} ∘ P_{7} ∘ Q_{5} ∘ P_{5}) ∘ (Q_{5} ∘ P_{5} ∘ Q_{3} ∘ P_{3}) ∘ Hom(P_{1}, Q′_{1}))(B),

which, since Q′_{1} ∘ B ∘ P_{1} is an element of the subspace V_{3} ∩ V_{5}, simplifies to

β · (Q_{7} ∘ P_{7})(Q′_{1} ∘ B ∘ P_{1}) = \frac{β}{2} · (Q′_{1} ∘ B ∘ P_{1} + K_{3}(Q′_{1} ∘ B ∘ P_{1})).

The inverse map

{A = K_{W} ∘ A ∘ K_{U} = K_{3}(A)} → Hom(U_{1}, W_{1})

46 1. ABSTRACT LINEAR ALGEBRA

is the composite:

(Hom(Id_{U_{1}}, P′_{1}) ∘ (Hom(Q_{1}, Id_{W}) ∘ Q_{3}) ∘ Q′_{3}) ∘ (\frac{2}{β} · P′_{5} ∘ Q′′′_{5})

= \frac{2}{β} · Hom(Q_{1}, P′_{1}) ∘ (Q_{5} ∘ Q′_{5}) ∘ P′_{5} ∘ (P_{5} ∘ Q_{5}) ∘ Q′′′_{5}
= \frac{2}{β} · Hom(Q_{1}, P′_{1}) ∘ (Q_{3} ∘ P_{3} ∘ Q_{5} ∘ P_{5}) ∘ Q_{5} ∘ Q′′′_{5}
= \frac{2}{β} · Hom(Q_{1}, P′_{1}) ∘ Q_{3} ∘ P_{3} ∘ Q_{5} ∘ Q′′′_{5},

which acts as A ↦ \frac{2}{β} · P′_{1} ∘ (\frac{1}{2} · (A + A ∘ K_{U})) ∘ Q_{1} = \frac{2}{β} · A ∘ Q_{1}.

CHAPTER 2 A Survey of Trace Elements

2.1. Endomorphisms: the scalar valued trace

In the following diagram, the canonical maps k_{V V}, e_{V V}, and f_{V V} are abbreviated k, e, f, and the double duality d_{V^{∗}⊗V} is abbreviated d.

Lemma 2.1. For any vector space V, the following diagram is commutative.

\xymatrix{ &  & (V^{∗} ⊗ V)^{∗} \\ End(V) \ar[urr]^{e} &  &  &  & End(V)^{∗} \ar[ull]^{k^{∗}} \\  & V^{∗} ⊗ V \ar[ul]^{k} \ar[uur]^{f} \ar[rr]^{d} &  & (V^{∗} ⊗ V)^{∗∗} \ar[uul]^{f^{∗}} \ar[ur]^{e^{∗}}}

Proof. The left triangle is commutative by the definition of f from Notation 1.65. The right triangle is commutative by Lemma 1.6, and the middle by Lemma 1.67.

The spaces End(V) and (V^{∗} ⊗ V)^{∗} each have the interesting property of containing a distinguished element, which is nonzero when V has nonzero dimension. The identity Id_{V} : v ↦ v is the distinguished element of End(V).

Definition 2.2. The distinguished element of (V^{∗} ⊗ V)^{∗} is the \underline{evaluation} operator, Ev_{V} : ϕ ⊗ v ↦ ϕ(v).

The two distinguished elements are related by e : Id_{V} ↦ Ev_{V}:

(2.1)  (e(Id_{V}))(ϕ ⊗ v) = ϕ(Id_{V} (v)) = Ev_{V} (ϕ ⊗ v).

Definition 2.3. For finite-dimensional V, define the \underline{trace} operator by

T r_{V} = (k^{∗})^{−1}(Ev_{V}) ∈ End(V)^{∗}.

This distinguished element is the output of either of the two previously mentioned distinguished elements under any path of maps in the above diagram leading to End(V)^{∗}, by Lemma 2.1, and the fact that all the arrows are invertible when V is finite-dimensional. At least one arrow in any path taking Id_{V} or Ev_{V} to T r_{V} is the inverse of one of the arrows indicated in the diagram.

Remark 2.4. Using Definition 2.3 as the definition of trace, so that T r_{V} (A) = Ev_{V} (k^{−1}(A)), is exactly the approach of [MB], [B] §II.4.3, and [K] §II.3. In [G_{2}] §I.8, this formula is stated as a consequence of a different definition of trace.

Lemma 2.5. For finite-dimensional V, and H ∈ End(V),

T r_{V^{∗}}(H^{∗}) = T r_{V} (H).

Proof. In this case, H^{∗} is t_{V V} (H). In the following diagram, t_{V V}, k_{V^{∗}V^{∗}}, e_{V^{∗}V^{∗}}, f_{V^{∗}V^{∗}}, and d_{V^{∗∗}⊗V^{∗}} are abbreviated t, k′, e′, f′, and d′. There is also a map p : V^{∗} ⊗ V → V^{∗∗} ⊗ V^{∗} from Notation 1.68, and p^{∗} maps the distinguished element Ev_{V^{∗}} ∈ (V^{∗∗} ⊗ V^{∗})^{∗} to Ev_{V}:

(p^{∗}(Ev_{V^{∗}}))(ϕ ⊗ v) = Ev_{V^{∗}}((d_{V} (v)) ⊗ ϕ) = (d_{V} (v))(ϕ) = ϕ(v) = Ev_{V} (ϕ ⊗ v).

[figure: (V^{∗} ⊗ V)^{∗}; End(V); e; t; End(V)^{∗}; k^{∗}; V^{∗} ⊗ V; k; f; d; p; (V^{∗} ⊗ V)^{∗∗}; f^{∗}; e^{∗}; p^{∗∗}; (V^{∗∗} ⊗ V^{∗})^{∗}; p^{∗}; End(V^{∗}); e′; End(V^{∗})^{∗}; k^{′∗}; t^{∗}; V^{∗∗} ⊗ V^{∗}; k′; f′; d′; (V^{∗∗} ⊗ V^{∗})^{∗∗}; f^{′∗}; e^{′∗}]

Some of the squares in the diagram are commutative, for example, k′ ∘ p = t ∘ k from Lemma 1.71, and then p^{∗} ∘ (k^{′∗}) = k^{∗} ∘ t^{∗} by Lemma 1.6, and this is enough to give the result:

t^{∗}(T r_{V^{∗}}) = (t^{∗} ∘ (k^{′∗})^{−1})(Ev_{V^{∗}}) = ((k^{∗})^{−1} ∘ p^{∗})(Ev_{V^{∗}}) = (k^{∗})^{−1}(Ev_{V}) = T r_{V} .

The equality q ∘ t = e from Lemma 1.54, which could fit in the back left square of the above diagram, shows that q maps the distinguished element Id_{V^{∗}} to Ev_{V}. So, there is another formula for the trace,

(2.2)  T r_{V} = ((k^{∗})^{−1} ∘ q)(Id_{V^{∗}}).

Lemma 2.6. For maps A : V → U and B : U → V between vector spaces of finite, but possibly different, dimensions, T r_{V} (B ∘ A) = T r_{U}(A ∘ B).

Proof. Abbreviated names for maps are used again in the following diagram, with primes in the lower pentagon.

[figure: (V^{∗} ⊗ V)^{∗}; [A^{∗}⊗B]^{∗}; End(V); e; _{_{Hom}_{(B, A)}}; End(V)^{∗}; k^{∗}; _{_{Hom}_{(A,B)^{∗}}}; V^{∗} ⊗ V; k; f; d; [B^{∗}⊗A]; [B^{∗}⊗A]^{∗}; (V^{∗} ⊗ V)^{∗∗}; f^{∗}; e^{∗}; [B^{∗}⊗A]^{∗∗}; (U^{∗} ⊗ U)^{∗}; End(U); _{_{Hom}_{(A,B)}}; e′; End(U)^{∗}; k^{′∗}; _{_{Hom}_{(B, A)^{∗}}}; U^{∗} ⊗ U; [A^{∗}⊗B]; k′; f′; d′; (U^{∗} ⊗ U)^{∗∗}; [A^{∗}⊗B]^{∗∗}; f^{′∗}; e^{′∗}]

Some of the squares in the diagram are commutative. In the back left square with upward arrows, it follows from Lemma 1.53 (and can also be checked directly) that:

e ∘ Hom(A, B) = [B^{∗} ⊗ A]^{∗} ∘ e′.

For a front left square, by Lemma 1.58,

Hom(B, A) ∘ k = k′ ∘ [B^{∗} ⊗ A],

then by Lemma 1.6, k^{∗}∘Hom(B, A)^{∗} = [B^{∗}⊗A]^{∗}∘k^{′∗}, corresponding to a back right square. The claimed equality follows from the following steps, including Lemma 2.1:

T r_{U}(A ∘ B) = T r_{U}(Hom(B, A)(Id_{V})) = (Hom(B, A)^{∗}(T r_{U}))(Id_{V})
= ((Hom(B, A)^{∗} ∘ (k^{′∗})^{−1} ∘ e′)(Id_{U}))(Id_{V})
= ((k^{∗})^{−1} ∘ [B^{∗} ⊗ A]^{∗} ∘ e′)(Id_{U}))(Id_{V})
= ((k^{∗})^{−1} ∘ e ∘ Hom(A, B))(Id_{U}))(Id_{V})
= ((e^{∗} ∘ d ∘ k^{−1} ∘ Hom(A, B))(Id_{U}))(Id_{V})
= (d(k^{−1}(Hom(A, B)(Id_{U}))))(e(Id_{V}))
= (e(Id_{V}))(k^{−1}(Hom(A, B)(Id_{U})))
= (((k^{∗})^{−1} ∘ e)(Id_{V}))(Hom(A, B)(Id_{U})) = T r_{V} (B ∘ A).

Example 2.7. In the case V = 𝕂, k(Id_{𝕂} ⊗ 1) = Id_{𝕂} ∈ End(𝕂) = 𝕂^{∗}. The trace is T r_{𝕂} = e^{∗}(d(Id_{𝕂} ⊗ 1)) ∈ 𝕂^{∗∗}, and for ϕ ∈ 𝕂^{∗}, T r_{𝕂}(ϕ) = (e(ϕ))(Id_{𝕂} ⊗ 1) = Id_{𝕂}(ϕ(1)) = ϕ(1). So, T r_{𝕂} = d_{𝕂}(1), and in particular, T r_{𝕂}(Id_{𝕂}) = 1.

Example 2.8. If V is finite-dimensional and admits a direct sum of the form V = 𝕂 ⊕ U, with projection P_{1} : V → 𝕂 and Q_{1} : 𝕂 → V, then by Lemma 2.6 and Example 2.7, T r_{V} (Q_{1} ∘ P_{1}) = T r_{𝕂}(P_{1} ∘ Q_{1}) = T r_{𝕂}(Id_{𝕂}) = 1. Similarly, if V is a direct sum of finitely many copies of 𝕂, V = 𝕂 ⊕ 𝕂 ⊕ ⋯ ⊕ 𝕂, then T r_{V} (Id_{V}) = T r_{V} (ΣQ_{i} ∘ P_{i}) = ΣT r_{𝕂}(P_{i} ∘ Q_{i}) = Σ1.

Example 2.9. Assume T r_{V} (Id_{V}) ≠ 0. Let End_{0}(V) denote the kernel of T r_{V}, i.e., the subspace of trace 0 endomorphisms. Recall from Lemma 1.96 that there exists a direct sum End(V) = 𝕂⊕End_{0}(V), and in particular, there exist constants α, β ∈ 𝕂 so that α · β · T r_{V} (Id_{V}) = 1, and a direct sum is defined by

P_{1}^{α} = α · T r_{V} ,
(2.3)  Q_{1}^{β} : 𝕂 → End(V) : γ ↦ β · γ · Id_{V} ,
P_{2} = Id_{End(V)} − Q_{1}^{β} ∘ P_{1}^{α},

and the subspace inclusion map Q_{2} : End_{0}(V) → End(V). Such a direct sum admits a free parameter and is generally not unique, but since Id_{V} is a canonical element of End(V) and is not in ker(T r_{V}) by assumption, Lemma 1.97 applies, and any choice of constants α, β leads to an equivalent direct sum. So, any endomorphism H can be written as a sum of a scalar multiple of Id_{V}, and a trace zero endomorphism:

H = \frac{T r_{V} (H)}{T r_{V} (Id_{V})} · Id_{V} + (H − \frac{T r_{V} (H)}{T r_{V} (Id_{V})} · Id_{V} ) ,

and this decomposition of H is canonical.

Theorem 2.10. For V finite-dimensional, and A ∈ End(V),

T r_{V} (A) = (Ev_{V} ∘ [Id_{V^{∗}} ⊗ A] ∘ k^{−1})(Id_{V}).

Proof. By Lemma 1.58,

[Id_{V^{∗}} ⊗ A] ∘ k^{−1} = [Id_{V}^{∗} ⊗ A] ∘ k^{−1} = k^{−1} ∘ Hom(Id_{V} , A),

so

(Ev_{V} ∘ [Id_{V^{∗}} ⊗ A] ∘ k^{−1})(Id_{V}) = Ev_{V} (k^{−1}(A)) = T r_{V} (A).

Remark 2.11. The idea of Theorem 2.10 (as in [K] §II.3) is that the trace of A is the output of the distinguished element k^{−1}(Id_{V}) under the composite of maps in this diagram:

\xymatrix{V^{∗} ⊗ V \ar[r]^{[Id_{V ^{∗}} ⊗A]} & V^{∗} ⊗ V \ar[r]^{Ev_{V}} & 𝕂.}

The statement of Theorem 2.10 could also be written as

T r_{V} (A) = ((d_{End(V)}(Id_{V})) ∘ Hom(k^{−1}, Ev_{V}) ∘ j)(Id_{V^{∗}} ⊗ A).

In terms of the scalar multiplication map l from Example 1.28 and the β = 1 case of (2.3) from Example 2.9,

(2.4)  Q_{1}^{1} : 𝕂 → End(V^{∗}) : 1 ↦ Id_{V^{∗}},

the composite [Q_{1}^{1} ⊗ Id_{End(V)}] ∘ l^{−1} : End(V) → End(V^{∗}) ⊗ End(V) takes A to Id_{V^{∗}} ⊗ A, so

(2.5)  T r_{V} = (d_{End(V)}(Id_{V})) ∘ Hom(k^{−1}, Ev_{V}) ∘ j ∘ [Q_{1}^{1} ⊗ Id_{End(V)}] ∘ l^{−1}.

The map Q_{1}^{1} from (2.4) is used in (2.5), and again in later Sections, without any assumption on the identity map’s trace, so Q_{1}^{1} is not necessarily part of the data for some direct sum as in Example 2.9.

Proposition 2.12. For V = V_{1} ⊕ V_{2}, A ∈ End(V_{1}), and B ∈ End(V_{2}), let A ⊕ B be the element of End(V) defined by A ⊕ B = Q_{1} ∘ A ∘ P_{1} + Q_{2} ∘ B ∘ P_{2}. If V is finite-dimensional, then

T r_{V} (A ⊕ B) = T r_{V_{1}}(A) + T r_{V_{2}}(B).

Proof. Recall V_{1} and V_{2} are finite-dimensional by Exercise 0.44. The construction of A ⊕ B is as in Lemma 1.82.

T r_{V} (A ⊕ B) = T r_{V} (Q_{1} ∘ A ∘ P_{1}) + T r_{V} (Q_{2} ∘ B ∘ P_{2})
= T r_{V_{1}}(P_{1} ∘ Q_{1} ∘ A) + T r_{V_{2}}(P_{2} ∘ Q_{2} ∘ B)
= T r_{V_{1}}(A) + T r_{V_{2}}(B),

using Lemma 2.6.

Proposition 2.13. If V is finite-dimensional, V = V_{1} ⊕ V_{2}, and K ∈ End(V), then

T r_{V} (K) = T r_{V_{1}}(P_{1} ∘ K ∘ Q_{1}) + T r_{V_{2}}(P_{2} ∘ K ∘ Q_{2}).

Proof. Using Lemma 2.6,

T r_{V} (K) = T r_{V} ((Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ K) = T r_{V_{1}}(P_{1} ∘ K ∘ Q_{1}) + T r_{V_{2}}(P_{2} ∘ K ∘ Q_{2}).

The formula T r_{V} (Id_{V}) = T r_{V_{1}}(Id_{V_{1}}) + T r_{V_{2}}(Id_{V_{2}}) can be considered as a special case of either Proposition 2.12 or Proposition 2.13.

Exercise 2.14. Given V finite-dimensional and A, P ∈ End(V), suppose P is an idempotent with image subspace V_{1} = P (V), and let P_{1} and Q_{1} be the projection and inclusion operators from Example 1.109. Then

T r_{V} (P ∘ A) = T r_{V_{1}}(P_{1} ∘ A ∘ Q_{1}).

In particular, for A = Id_{V}, T r_{V} (P) = T r_{V_{1}}(Id_{V_{1}}).

Exercise 2.15. ([AFMC] P1993-3) For V finite-dimensional, ϕ ∈ V^{∗}, v ∈ V, the following are equivalent.

(1) T r_{V} (k_{V V} (ϕ ⊗ v)) = 1.

(2) k_{V V} (ϕ ⊗ v) ∈ End(V) is a non-zero idempotent.

Proposition 2.16. ([G_{1}] §IV.7) For V finite-dimensional and A ∈ End(V), the following are equivalent.

(1) T r_{V} (A ∘ B) = 0 for all B.

(2) A = 0_{End(V)}.

Proof.

Hom(Id_{V} , A)^{∗}(T r_{V}) = (Hom(Id_{V} , A)^{∗} ∘ (k^{∗})^{−1} ∘ e)(Id_{V})
= ((k^{∗})^{−1} ∘ e ∘ Hom(A, Id_{V}))(Id_{V})
= ((k^{∗})^{−1} ∘ e)(A),

by the commutativity of the diagram from Lemma 2.6, with U = V. If

T r_{V} (A ∘ B) = (Hom(Id_{V} , A)^{∗}(T r_{V}))(B) = (((k^{∗})^{−1} ∘ e)(A))(B)

is always zero, then ((k^{∗})^{−1} ∘e)(A) = 0_{End(V)^{∗}}, and since (k^{∗})^{−1} ∘e has zero kernel, A must be 0_{End(V)}. The converse is trivial.

Proposition 2.17. ([B] §II.10.11) Suppose V is finite-dimensional and Φ ∈ End(V)^{∗}. Then there exists F ∈ End(V) such that Φ(A) = T r_{V} (F ∘ A) for all A ∈ End(V).

Proof. Let F = e^{−1}(k^{∗}(Φ)). The result follows from the commutativity of the appropriate paths in the diagram for the Proof of Lemma 2.6 in the case U = V.

Φ(A) = (Hom(Id_{V} , A)^{∗}(Φ))(Id_{V})
= ((e^{∗} ∘ d ∘ k^{−1} ∘ Hom(A, Id_{V}) ∘ e^{−1} ∘ k^{∗})(Φ))(Id_{V})
= ((k^{−1})^{∗}(e(Id_{V})))(Hom(A, Id_{V})(e^{−1}(k^{∗}(Φ))))
= T r_{V} (F ∘ A).

Proposition 2.18. ([B] §II.10.11) For V finite-dimensional and Φ ∈ End(V)^{∗}, the following are equivalent.

(1) Φ satisfies Φ(A ∘ B) = Φ(B ∘ A) for all A, B ∈ End(V).

(2) There exists λ ∈ 𝕂 such that Φ = λ · T r_{V}.

Proof. By Proposition 2.17, Φ(A ∘ B) = T r_{V} (F ∘ A ∘ B) = Φ(B ∘ A) = T r_{V} (F ∘ B ∘ A). By Lemma 2.6, T r_{V} (F ∘ B ∘ A) = T r_{V} (A ∘ B ∘ F) for all B, so Hom(A, F)^{∗}(T r_{V}) = Hom(F, A)^{∗}(T r_{V}). Then

Hom(A, F)^{∗}((k^{∗})^{−1}(e(Id_{V}))) = Hom(F, A)^{∗}((k^{∗})^{−1}(e(Id_{V})))
(k^{∗})^{−1}(e(Hom(F, A)(Id_{V}))) = (k^{∗})^{−1}(e(Hom(A, F)(Id_{V})))
((k^{∗})^{−1} ∘ e)(A ∘ F) = ((k^{∗})^{−1} ∘ e)(F ∘ A),

so A ∘ F = F ∘ A for all A, and so F = λ · Id_{V} by Claim 0.51. The converse follows from Lemma 2.6.

Proposition 2.19. ([G_{1}] §IV.7) Suppose T r_{V} (Id_{V}) ≠ 0, and that the operator Ω ∈ End(End(V)) satisfies Ω(A∘B) = (Ω(A))∘(Ω(B)) for all A, B, and Ω(Id_{V}) = Id_{V}. Then T r_{V} (Ω(H)) = T r_{V} (H) for all H ∈ End(V).

Proof.

T r_{V} (Ω(A ∘ B)) = T r_{V} ((Ω(A)) ∘ (Ω(B))))
= T r_{V} ((Ω(B)) ∘ (Ω(A)))
= T r_{V} (Ω(B ∘ A)),

so Proposition 2.18 applies to Ω^{∗}(T r_{V}) and T r_{V} (Ω(H)) = λ · T r_{V} (H). The second property of Ω implies T r_{V} (Ω(Id_{V})) = T r_{V} (Id_{V}) = λ · T r_{V} (Id_{V}), so either T r_{V} (Id_{V}) = 0, or λ = 1 and T r_{V} (Ω(H)) = T r_{V} (H) for all H.

Remark 2.20. The following Lemma is a generalization of Example 2.7, motivated by a property of a “line bundle.”

Lemma 2.21. Suppose that L is a finite-dimensional vector space, and that the evaluation map Ev_{L} : L^{∗} ⊗ L → 𝕂 is invertible. Then T r_{L} : End(L) → 𝕂 is invertible and T r_{L}(Id_{L}) = 1.

Proof. k = k_{LL} and e = e_{LL} are invertible. Ev_{L} ≠ 0_{(L^{∗}⊗L)^{∗}}, so Id_{L} = e^{−1}(Ev_{L}) ≠ 0_{End(L)}. T r_{L} = Ev_{L} ∘k^{−1} is invertible, so T r_{L}(Id_{L}) ≠ 0, and Example 2.9 applies. In particular, there is some β ∈ 𝕂 so that β · T r_{L}(Id_{L}) = 1, and a map Q_{1}^{β} : 𝕂 → End(L) : γ ↦ β · γ · Id_{L} as in Equation (2.3) so that T r_{L} ∘ Q_{1}^{β} = Id_{𝕂}. It follows that

(Ev_{L} ∘ k^{−1}) ∘ Q_{1}^{β} = Id_{𝕂} ⟹ Q_{1}^{β} ∘ Ev_{L} = k.

There is also some v ∈ L, and there is some ϕ ∈ L^{∗}, so that Ev_{L}(ϕ ⊗ v) ≠ 0, and so ϕ(v) ≠ 0, and v ≠ 0_{L}. Then,

k(ϕ ⊗ v) = (Q_{1}^{β} ∘ Ev_{L})(ϕ ⊗ v) = β · ϕ(v) · Id_{L},

so

(k(ϕ ⊗ v))(v) = ϕ(v) · v = (β · ϕ(v) · Id_{L})(v) = β · ϕ(v) · v,

which implies β = 1.

Remark 2.22. The following Proposition is proved in a different way by [AS^{2}].

Proposition 2.23. Given V finite-dimensional and any positive integer ν, if P_{1} : V → V and P_{2} : V → V are any idempotents, then

T r_{V} ((P_{1} − P_{2})^{2ν+1}) = T r_{V} (P_{1} − P_{2}).

Proof. The odd power refers to a composite (P_{1} − P_{2}) ∘ ⋯ ∘ (P_{1} − P_{2}). It can be shown by induction on ν that there exist constants α_{ν,ι} ∈ 𝕂, ι = 1, …, ν, so that the composite expands:

(P_{1} − P_{2})^{2ν+1} = P_{1} − P_{2} + ∑_{ι=1}^{ν} α_{ν,ι} · ((P_{1} ∘ P_{2})^{ι} ∘ P_{1} − (P_{2} ∘ P_{1})^{ι} ∘ P_{2}).

The claim then follows from Lemma 2.6.

2.2. The generalized trace

An analogue of the trace T r_{V} : End(V) → 𝕂 is the \underline{generalized trace}, a map

T r_{V;U,W} : Hom(V ⊗ U, V ⊗ W) → Hom(U, W),

constructed in Definition 2.24.

2.2.1. Defining the generalized trace.

The following particular cases of the canonical j maps will be used repeatedly:

j_{1} : End(V)^{∗}⊗ End(Hom(U,W)) → Hom(End(V) ⊗ Hom(U,W), 𝕂 ⊗ Hom(U,W))
j_{2} : End(V) ⊗ Hom(U, W) → Hom(V ⊗ U, V ⊗ W).

If V is finite-dimensional, then both j_{1} and j_{2} are invertible, by Claim 1.34. Denote by l_{1} the scalar multiplication map 𝕂 ⊗ Hom(U, W) → Hom(U, W). The domain of j_{1} contains the distinguished element T r_{V} ⊗ Id_{Hom(U,W)}.

Definition 2.24. For finite-dimensional V, define

T r_{V;U,W} = (Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)}).

Note that the finite-dimensionality of V is used in the Definition, since j_{2} must be invertible, but U and W may be arbitrary vector spaces.

Example 2.25. A map of the form j_{2}(A⊗B) : V ⊗U → V ⊗W, for A : V → V and B : U → W, has trace

T r_{V;U,W} (j_{2}(A ⊗ B)) = l_{1}((j_{1}(T r_{V} ⊗ Id_{Hom(U,W)}))(A ⊗ B)) = T r_{V} (A) · B.

In the V = 𝕂 case, the trace is an invertible map. Denote scalar multiplication maps l_{W} : 𝕂 ⊗ W → W and l_{U} : 𝕂 ⊗ U → U.

Theorem 2.26. For any vector spaces U, W,

T r_{𝕂;U,W} = Hom(l_{U}^{−1}, l_{W}) : Hom(𝕂 ⊗ U, 𝕂 ⊗ W) → Hom(U, W).

Proof. For any ϕ ∈ 𝕂^{∗}, F ∈ Hom(U, W), the equation (ϕ(1) · F) ∘ l_{U} = l_{W} ∘ [ϕ ⊗ F] from Lemma 1.38 can be rewritten using j_{2} : 𝕂^{∗} ⊗ Hom(U, W) → Hom(𝕂 ⊗ U, 𝕂 ⊗ W),

Hom(l_{U}^{−1}, l_{W}) ∘ j_{2} = l_{1} ∘ (j_{1}((d_{𝕂}(1)) ⊗ Id_{Hom(U,W)})) : ϕ ⊗ F ↦ ϕ(1) · F.

The equality follows from Example 2.7, where T r_{𝕂} = d_{𝕂}(1):

Hom(l_{U}^{−1}, l_{W}) = l_{1} ∘ (j_{1}((d_{𝕂}(1)) ⊗ Id_{Hom(U,W)})) ∘ j_{2}^{−1} = T r_{𝕂;U,W} .

In the U = W = 𝕂 case, the generalized trace is related to the scalar trace as in the following Theorem.

Theorem 2.27. For finite-dimensional V, the scalar multiplication map l_{V} : V ⊗ 𝕂 → V, and any H ∈ End(V),

(T r_{V;𝕂,𝕂}(l_{V}^{−1} ∘ H ∘ l_{V}))(1) = T r_{V} (H).

Equivalently,

(2.6)  T r_{V;𝕂,𝕂}(l_{V}^{−1} ∘ H ∘ l_{V}) = T r_{V} (H) · Id_{𝕂}.

Proof. In the following diagram,

\xymatrix{End(V)^{∗} ⊗ End(𝕂^{∗}) \ar[r]^{a_{1}} \ar[d]^{j_{1}} & End(V)^{∗}⊗ Hom(𝕂, 𝕂^{∗}) \ar[d]^{j_{3}} & End(V)^{∗} ⊗ 𝕂^{∗} \ar[ddd]_{k_{End(V ),𝕂^{∗}}} \ar[l]_{a_{5}} \\ Hom(End(V)⊗𝕂^{∗},𝕂⊗𝕂^{∗}) \ar[r]^{a_{2}} \ar[d]^{Hom(Id_{End(V )⊗𝕂^{∗}} ,l_{1})} & Hom(End(V)⊗𝕂,𝕂⊗𝕂^{∗}) \ar[d]^{Hom(Id_{End(V )⊗𝕂},l_{1})} \\ Hom(End(V) ⊗ 𝕂^{∗}, 𝕂^{∗}) \ar[r]^{a_{3}} & Hom(End(V) ⊗ 𝕂, 𝕂^{∗}) \\ Hom(End(V ⊗𝕂), 𝕂^{∗}) \ar[u]_{Hom(j_{2},Id_{𝕂^{∗}})} & Hom(Hom(V⊗𝕂,V),𝕂^{∗}) \ar[r]^{a_{6}} \ar[l]_{a_{4}} & Hom(End(V),𝕂^{∗}) \ar[ul]_{Hom(l_{2},Id_{𝕂^{∗}})}}

the horizontal arrows are

a_{1} = [Id_{End(V)^{∗}} ⊗ Hom(m, Id_{𝕂^{∗}})]
a_{2} = Hom([Id_{End(V)} ⊗ m], Id_{𝕂⊗𝕂^{∗}})
a_{3} = Hom([Id_{End(V)} ⊗ m], Id_{𝕂^{∗}})
a_{4} = Hom(Hom(Id_{V ⊗𝕂}, l_{V}), Id_{𝕂^{∗}})
a_{5} = [Id_{End(V)^{∗}} ⊗ Hom(Id_{𝕂}, m)]
a_{6} = Hom(Hom(l_{V} , Id_{V}), Id_{𝕂^{∗}}),

for m : 𝕂 → 𝕂^{∗} as in Definition 1.20, so that m(α) : λ ↦ λ · α. So there is not much going on besides scalar multiplication, including the map l_{2} : End(V) ⊗ 𝕂 → End(V).

Starting with the element T r_{V} ⊗ Id_{𝕂^{∗}} in the space in the upper left corner, its output under the composite map going downward and then right to the lower right corner is, using Definition 2.24, T r_{V;𝕂,𝕂} ∘ Hom(l_{V} , l_{V}^{−1}), as in the LHS of the claim (2.6). The output in the path going right and then downward is

k_{End(V),𝕂^{∗}}(T r_{V} ⊗ (m^{−1} ∘ Id_{𝕂^{∗}} ∘ m)) = k_{End(V),𝕂^{∗}}(T r_{V} ⊗ Id_{𝕂}) : H ↦ T r_{V} (H) · Id_{𝕂},

corresponding to the RHS of the claim. The claimed equality follows from just the commutativity of the diagram, without using any special properties of the trace T r_{V}.

The upper left block is commutative by Lemma 1.37 and the middle left block is commutative by Lemma 1.6. For the commutativity of the lower block, it is enough, by Lemma 1.6, to check this equality, for A ∈ End(V), α, β ∈ 𝕂, v ∈ V:

Hom(Id_{V ⊗𝕂}, l_{V}) ∘ j_{2} ∘ [Id_{End(V)} ⊗ m] :
A ⊗ α ↦ l_{V} ∘ [A ⊗ (m(α))] : v ⊗ β ↦ l_{V} ((A(v)) ⊗ (β · α)) = β · α · A(v),
Hom(l_{V} , Id_{V}) ∘ l_{2} :
A ⊗ α ↦ (α · A) ∘ l_{V} : v ⊗ β ↦ (α · A)(β · v).

Checking the commutativity of the block on the right, for Φ ∈ End(V)^{∗}, ϕ ∈ 𝕂^{∗},

Hom(Id_{End(V)⊗𝕂}, l_{1}) ∘ j_{3} ∘ [Id_{End(V)^{∗}} ⊗ Hom(Id_{𝕂}, m)] :
Φ ⊗ ϕ ↦ l_{1} ∘ [Φ ⊗ (m ∘ ϕ)] :
A ⊗ α ↦ (Φ(A)) · (m(ϕ(α))) : β ↦ (Φ(A)) · β · ϕ(α),
Hom(l_{2}, Id_{𝕂^{∗}}) ∘ k_{End(V),𝕂^{∗}} :
Φ ⊗ ϕ ↦ (k_{End(V),𝕂^{∗}}(Φ ⊗ ϕ)) ∘ l_{2} :
A ⊗ α ↦ (k_{End(V),𝕂^{∗}}(Φ ⊗ ϕ))(α · A) = (Φ(α · A)) · ϕ : β ↦ (Φ(α · A)) · ϕ(β).

2.2.2. Properties of the generalized trace.

The next Theorems in this Section are straightforward linear algebra identities for the generalized trace.

Remark 2.28. Versions of some of the results in this Section are stated in a more general context of category theory, and given different proofs, in [Maltsiniotis] §3.5 or [JSV] §2. The result of Theorem 2.26 is related to a property called “vanishing” by [JSV], and Theorem 2.29 and Theorem 2.30 are “naturality” properties ([JSV]).

An analogue of Lemma 2.6 applies to maps A : V → V′ and B : V′ ⊗ U → V ⊗ W, using the canonical maps

j_{U} : Hom(V, V′) ⊗ End(U) → Hom(V ⊗ U, V′ ⊗ U),
j_{W} : Hom(V, V′) ⊗ End(W) → Hom(V ⊗ W, V′ ⊗ W).

Theorem 2.29. For finite-dimensional V, V′,

T r_{V;U,W} (B ∘ (j_{U}(A ⊗ Id_{U}))) = T r_{V′;U,W} ((j_{W} (A ⊗ Id_{W})) ∘ B).

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j_{1}} & M_{12} \ar[d]^{j′′_{1}} & M_{13} \ar[l]_{a_{2}} \ar[d]^{j′_{1}} \\ M_{21} \ar[r]^{a_{3}} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & M_{22} \ar[d]^{Hom((j′′_{2})^{−1},l_{1})} & M_{23} \ar[l]_{a_{4}} \ar[d]^{Hom((j′_{2})^{−1},l_{1})} \\ M_{31} \ar[u]^{Hom(j_{2},l_{1}^{−1})} \ar[r]^{a_{5}} & M_{32} \ar[u]^{Hom(j′′_{2},l_{1}^{−1})} & M_{33} \ar[u]^{Hom(j′_{2},l_{1}^{−1})} \ar[l]_{a_{6}}}

the objects are

M_{11} = End(V)^{∗} ⊗ End(Hom(U, W))
M_{21} = Hom(End(V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U, W))
M_{31} = Hom(Hom(V ⊗ U, V ⊗ W), Hom(U, W))
M_{12} = Hom(V′, V)^{∗} ⊗ End(Hom(U, W))
M_{22} = Hom(Hom(V′, V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U, W))
M_{32} = Hom(Hom(V′ ⊗ U, V ⊗ W), Hom(U, W))
M_{13} = End(V′)^{∗} ⊗ End(Hom(U, W))
M_{23} = Hom(End(V′) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U, W))
M_{33} = Hom(Hom(V′ ⊗ U, V′ ⊗ W), Hom(U, W)),

where the left and right columns are the maps from the definition of trace and the horizontal arrows in the diagram are

a_{1} = [Hom(A, Id_{V})^{∗} ⊗ Id_{End(Hom(U,W))}]
a_{2} = [Hom(Id_{V′}, A)^{∗} ⊗ Id_{End(Hom(U,W))}]
a_{3} = Hom([Hom(A, Id_{V}) ⊗ Id_{Hom(U,W)}], Id_{𝕂⊗Hom(U,W)})
a_{4} = Hom([Hom(Id_{V′}, A) ⊗ Id_{Hom(U,W)}], Id_{𝕂⊗Hom(U,W)})
a_{5} = Hom(Hom(j_{U}(A ⊗ Id_{U}), Id_{V ⊗W}), Id_{Hom(U,W)})
a_{6} = Hom(Hom(Id_{V′⊗U}, j_{W} (A ⊗ Id_{W})), Id_{Hom(U,W)}).

The two quantities in the statement of the Theorem are

T r_{V;U,W} (B ∘ (j_{U}(A ⊗ Id_{U}))) = (a_{5}(T r_{V;U,W}))(B),
T r_{V′;U,W} ((j_{W} (A ⊗ Id_{W})) ∘ B) = (a_{6}(T r_{V′;U,W}))(B).

Each of the squares in the diagram is commutative, by Lemma 1.6 and Lemma 1.37. By Lemma 2.6, Hom(A, Id_{V})^{∗}(T r_{V}) = Hom(Id_{V′}, A)^{∗}(T r_{V′}), so

a_{1}(T r_{V} ⊗ Id_{Hom(U,W)}) = (Hom(A, Id_{V})^{∗}(T r_{V})) ⊗ Id_{Hom(U,W)}
= (Hom(Id_{V′}, A)^{∗}(T r_{V′})) ⊗ Id_{Hom(U,W)}
= a_{2}(T r_{V′} ⊗ Id_{Hom(U,W)}).

The Theorem follows from the commutativity of the diagram:

a_{5}(T r_{V;U,W}) = (a_{5} ∘ Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom((j′′_{2})^{−1}, l_{1}) ∘ j′′_{1} ∘ a_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom((j′′_{2})^{−1}, l_{1}) ∘ j′′_{1} ∘ a_{2})(T r_{V′} ⊗ Id_{Hom(U,W)})
= (a_{6} ∘ Hom((j′_{2})^{−1}, l_{1}) ∘ j′_{1})(T r_{V′} ⊗ Id_{Hom(U,W)})
= a_{6}(T r_{V′;U,W}).

The general strategy for the preceding proof will be repeated in some subsequent proofs. To derive an equality involving the generalized trace, a diagram is set up with the maps from Definition 2.24 on the left and right. The lowest row will be the desired theorem, and the top row is the “key step,” which is either obvious, or which uses the previously derived properties of the scalar valued trace. There will be little choice in selecting canonical maps as horizontal arrows, and the commutativity of the diagram will give the theorem as a consequence of the key step. We remark that the canonical maps m, n, and q will not be needed until Section 2.3.

Theorem 2.30. If V is finite-dimensional, then for any maps A : V ⊗ U → V ⊗ W, B : W → W′, and C : U′ → U, the composite [Id_{V} ⊗ B] ∘ A ∘ [Id_{V} ⊗ C] : V ⊗ U′ → V ⊗ W′ has trace

T r_{V;U′,W′}([Id_{V} ⊗ B] ∘ A ∘ [Id_{V} ⊗ C]) = B ∘ (T r_{V;U,W} (A)) ∘ C.

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j_{1}} & M_{12} \ar[d]^{j′′_{1}} & M_{13} \ar[l]_{a_{2}} \ar[d]^{j′_{1}} \\ M_{21} \ar[r]^{a_{3}} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & M_{22} \ar[d]^{Hom(j_{2}^{−1},l′_{1})} & M_{23} \ar[l]_{a_{4}} \ar[d]^{Hom((j′_{2})^{−1},l′_{1})} \\ M_{31} \ar[u]^{Hom(j_{2},l_{1}^{−1})} \ar[r]^{a_{5}} & M_{32} \ar[u]^{Hom(j_{2},(l′_{1})^{−1})} & M_{33} \ar[u]^{Hom(j′_{2},(l′_{1})^{−1})} \ar[l]_{a_{6}}}

the objects are

M_{11} = End(V)^{∗} ⊗ End(Hom(U, W))
M_{21} = Hom(End(V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U, W))
M_{31} = Hom(Hom(V ⊗ U, V ⊗ W), Hom(U, W))
M_{12} = End(V)^{∗} ⊗ Hom(Hom(U, W), Hom(U′, W′))
M_{22} = Hom(End(V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U′, W′))
M_{32} = Hom(Hom(V ⊗ U, V ⊗ W), Hom(U′, W′))
M_{13} = End(V)^{∗} ⊗ End(Hom(U′, W′))
M_{23} = Hom(End(V) ⊗ Hom(U′, W′), 𝕂 ⊗ Hom(U′, W′))
M_{33} = Hom(Hom(V ⊗ U′, V ⊗ W′), Hom(U′, W′)),

where the left and right columns are the maps from the definition of trace and the horizontal arrows in the diagram are

a_{1} = [Id_{End(V)^{∗}} ⊗ Hom(Id_{Hom(U,W)}, Hom(C, B))]
a_{2} = [Id_{End(V)^{∗}} ⊗ Hom(Hom(C, B), Id_{Hom(U′,W′)})]
a_{3} = Hom(Id_{End(V)⊗Hom(U,W)}, [Id_{𝕂} ⊗ Hom(C, B)])
a_{4} = Hom([Id_{End(V)} ⊗ Hom(C, B)], Id_{𝕂⊗Hom(U′,W′)})
a_{5} = Hom(Id_{Hom(V ⊗U,V ⊗W)}, Hom(C, B))
a_{6} = Hom(Hom([Id_{V} ⊗ C], [Id_{V} ⊗ B]), Id_{Hom(U′,W′)}).

The two quantities in the statement of the Theorem are

T r_{V;U′,W′}([Id_{V} ⊗ B] ∘ A ∘ [Id_{V} ⊗ C]) = (a_{6}(T r_{V;U′,W′}))(A)
B ∘ (T r_{V;U,W} (A)) ∘ C = (a_{5}(T r_{V;U,W}))(A).

Squares except the lower left commute by Lemma 1.6 and Lemma 1.37; for the remaining square, let λ ∈ 𝕂, E ∈ Hom(U, W):

(l′_{1} ∘ [Id_{𝕂} ⊗ Hom(C, B)])(λ ⊗ E) = l′_{1}(λ ⊗ (B ∘ E ∘ C)) = λ · (B ∘ E ∘ C)
(Hom(C, B) ∘ l_{1})(λ ⊗ E) = B ∘ (λ · E) ∘ C = λ · (B ∘ E ∘ C).

The “key step” uses a property of the identity map, and not any properties of the trace:

a_{1}(T r_{V} ⊗ Id_{Hom(U,W)}) = T r_{V} ⊗ Hom(C, B) = a_{2}(T r_{V} ⊗ Id_{Hom(U′,W′)})

The Theorem follows from the commutativity of the diagram:

a_{5}(T r_{V;U,W}) = (a_{5} ∘ Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom(j_{2}^{−1}, l′_{1}) ∘ j′′_{1} ∘ a_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom(j_{2}^{−1}, l′_{1}) ∘ j′′_{1} ∘ a_{2})(T r_{V} ⊗ Id_{Hom(U′,W′)})
= (a_{6} ∘ Hom((j′_{2})^{−1}, l′_{1}) ∘ j′_{1})(T r_{V} ⊗ Id_{Hom(U′,W′)})
= a_{6}(T r_{V;U′,W′}).

Corollary 2.31. If V and V′ are finite-dimensional then for any maps A : V → V′, B : W → W′, C : U′ → U, the following diagram is commutative.

\xymatrix{Hom(V ⊗ U, V ⊗ W) \ar[r]^{T r_{V ;U,W}} & Hom(U, W) \ar[dd]^{Hom(C,B)} \\ Hom(V′ ⊗ U, V ⊗ W) \ar[u]^{Hom([A⊗Id_{U}],Id_{V ⊗W})} \ar[d]_{Hom([Id_{V ′} ⊗C],[A⊗B])} \\ Hom(V′ ⊗ U′, V′ ⊗ W′) \ar[r]^{T r_{V ′;U′,W′}} & Hom(U′, W′)}

Proof. This follows from Theorem 2.29 and Theorem 2.30.

Lemma 2.32. The following diagram is commutative.

\xymatrix{V_{1}^{∗} ⊗ V_{2}^{∗} ⊗ W_{1} ⊗ W_{2} \ar[r]^{s} \ar[d]^{[j⊗Id_{W_{1}⊗W_{2}} ]} & V_{1}^{∗} ⊗ W_{1} ⊗ V_{2}^{∗} ⊗ W_{2} \ar[d]^{[k_{V_{1}W_{1}} ⊗k_{V_{2}W_{2}} ]} \\ Hom(V_{1} ⊗ V_{2}, 𝕂 ⊗ 𝕂) ⊗ W_{1} ⊗ W_{2} \ar[d]^{[Hom(Id_{V_{1}⊗V_{2}} ,l)⊗Id_{W_{1}⊗W_{2}} ]} & Hom(V_{1}, W_{1}) ⊗ Hom(V_{2}, W_{2}) \ar[d]^{j} \\ (V_{1} ⊗ V_{2})^{∗} ⊗ W_{1} ⊗ W_{2} \ar[r]^{k_{V_{1}⊗V_{2},W_{1}⊗W_{2}}} & Hom(V_{1} ⊗ V_{2}, W_{1} ⊗ W_{2})}

Proof. The map s switches the middle two factors of the tensor product (as in Example 1.29), and l is multiplication of elements of 𝕂.

ϕ_{1} ⊗ ϕ_{2} ⊗ w_{1} ⊗ w_{2} ↦ (j ∘ [k_{V_{1}W_{1}} ⊗ k_{V_{2}W_{2}}] ∘ s)(ϕ_{1} ⊗ ϕ_{2} ⊗ w_{1} ⊗ w_{2})
= [(k_{V_{1}W_{1}}(ϕ_{1} ⊗ w_{1})) ⊗ (k_{V_{2}W_{2}}(ϕ_{2} ⊗ w_{2}))] :
v_{1} ⊗ v_{2} ↦ ((ϕ(v_{1})) · w_{1}) ⊗ ((ϕ_{2}(v_{2})) · w_{2}),
ϕ_{1} ⊗ ϕ_{2} ⊗ w_{1} ⊗ w_{2} ↦ k_{V_{1}⊗V_{2},W_{1}⊗W_{2}}((l ∘ [ϕ_{1} ⊗ ϕ_{2}]) ⊗ w_{1} ⊗ w_{2}) :
v_{1} ⊗ v_{2} ↦ ϕ_{1}(v_{1}) · ϕ_{2}(v_{2}) · w_{1} ⊗ w_{2}.

Remark 2.33. The above result appears in [K] §II.2, and is related to a matrix algebra equation in [Magnus] §3.6.

Theorem 2.34. For finite-dimensional V and U, and A : V ⊗ U → V ⊗ U,

T r_{U}(T r_{V;U,U}(A)) = T r_{V ⊗U}(A).

Proof. As in Lemma 2.6, the maps k_{V V} and k_{UU} are abbreviated k and k′, and the corresponding map for V ⊗U is denoted k′′ : (V ⊗U)^{∗}⊗V ⊗U → End(V ⊗U).

By Lemma 2.32, these k maps are related by the following commutative diagram.

\xymatrix{V^{∗} ⊗ V ⊗ U^{∗} ⊗ U \ar[r]^{[k⊗k′]} \ar[d]^{s} & End(V) ⊗ End(U) \ar[ddd]^{j_{2}} \\ V^{∗} ⊗ U^{∗} ⊗ V ⊗ U \ar[d]^{[j⊗Id_{V ⊗U}]} \\ Hom(V ⊗ U, 𝕂 ⊗ 𝕂) ⊗ V ⊗ U \ar[d]^{[Hom(Id_{V ⊗U} ,l)⊗Id_{V ⊗U}]} \\ (V ⊗ U)^{∗} ⊗ V ⊗ U \ar[r]^{k′′} & End(V ⊗ U)}

The composite of maps in the left column is abbreviated a_{1}. In particular, since U and V are assumed finite-dimensional, all the arrows in the square are invertible. In the following diagram,

\xymatrix{End(V)^{∗} ⊗ End(End(U)) \ar[r]^{a_{2}} \ar[d]^{j_{1}} & (V^{∗} ⊗ V)^{∗} ⊗ End(U^{∗} ⊗ U) \ar[d]^{j′_{1}} \ar[l]^{a_{2}^{−1}} \\ Hom(End(V)⊗End(U), 𝕂⊗End(U)) \ar[r]^{a_{3}} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & Hom(V^{∗}⊗V ⊗U^{∗}⊗U, 𝕂⊗U^{∗}⊗U) \ar[d]^{Hom(a_{1}^{−1},k′∘l′_{1})} \\ Hom(End(V ⊗ U), End(U)) \ar[r]^{a_{4}} \ar[u]^{Hom(j_{2},l_{1}^{−1})} \ar[d]^{Hom(Id_{End(V⊗U)},T r_{U})} & Hom((V ⊗ U)^{∗} ⊗ V ⊗ U, End(U)) \ar[u]^{Hom(a_{1},(l′_{1})^{−1}∘(k′)^{−1})} \ar[d]_{Hom(Id_{(V⊗U)^{∗}⊗V⊗U} ,T r_{U})} \\ End(V ⊗ U)^{∗} \ar[r]^{k^{′′∗}} & ((V ⊗ U)^{∗} ⊗ V ⊗ U)^{∗}}

the horizontal arrows are

a_{2} = [k^{∗} ⊗ Hom(k′, (k′)^{−1})]
a_{2}^{−1} = [(k^{∗})^{−1} ⊗ Hom((k′)^{−1}, k′)]
a_{3} = Hom([k ⊗ k′], [Id_{𝕂} ⊗ (k′)^{−1}])
a_{4} = Hom(k′′, Id_{End(U)}),

and the statement of the Theorem is that

Hom(Id_{End(V ⊗U)}, T r_{U})(T r_{V;U,U}) = T r_{V ⊗U}.

The upper square is commutative by Lemma 1.37, and Lemma 1.6 applies easily to the commutativity of the lower square, and to that of the middle square using k′′ ∘ a_{1} = j_{2} ∘ [k ⊗ k′], from the first diagram, and l_{1} ∘ [Id_{𝕂} ⊗ k′] = k′ ∘ l′_{1}, by Lemma 1.38.

The commutativity of this square,

\xymatrix{V^{∗} ⊗ V ⊗ U^{∗} ⊗ U \ar[r]^{l′_{1}∘(j′_{1}(Ev_{V} ⊗Id_{U^{∗}⊗U}))} \ar[d]^{a_{1}} & U^{∗} ⊗ U \ar[d]^{Ev_{U}} \\ (V ⊗ U)^{∗} ⊗ V ⊗ U \ar[r]^{Ev_{V ⊗U}} & 𝕂}

(Ev_{V ⊗U} ∘ a_{1})(ϕ ⊗ v ⊗ ξ ⊗ u) = Ev_{V ⊗U}((l∘[ϕ ⊗ ξ]) ⊗ v ⊗ u)
= ϕ(v) · ξ(u)
(Ev_{U} ∘ l′_{1} ∘ (j′_{1}(Ev_{V} ⊗ Id_{U^{∗}⊗U})))(ϕ ⊗ v ⊗ ξ ⊗ u) = Ev_{U}(ϕ(v) · ξ ⊗ u)
= ϕ(v) · ξ(u),

implies that the distinguished elements Ev_{V} ⊗ Id_{U^{∗}⊗U} and Ev_{V ⊗U} are related by the right column of maps in the second diagram:

Ev_{V} ⊗ Id_{U^{∗}⊗U} ↦ T r_{U} ∘ k′ ∘ l′_{1} ∘ (j′_{1}(Ev_{V} ⊗ Id_{U^{∗}⊗U})) ∘ a_{1}^{−1}
= Ev_{U} ∘ l′_{1} ∘ (j′_{1}(Ev_{V} ⊗ Id_{U^{∗}⊗U})) ∘ a_{1}^{−1}
= Ev_{V ⊗U}.

The above equation used the definition of T r_{U}. Along the top row, the key step uses the definition of T r_{V}:

a_{2}^{−1}(Ev_{V} ⊗ Id_{U^{∗}⊗U}) = ((k^{∗})^{−1}(Ev_{V})) ⊗ (k′ ∘ Id_{U^{∗}⊗U} ∘ (k′)^{−1}) = T r_{V} ⊗ Id_{End(U)}.

The Theorem follows:

T r_{U} ∘ T r_{V;U,U} = (Hom(Id_{End(V ⊗U)}, T r_{U}) ∘ Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{End(U)})
= (Hom(Id_{End(V ⊗U)}, T r_{U})∘Hom(j_{2}^{−1}, l_{1}) ∘j_{1}∘a_{2}^{−1})(Ev_{V} ⊗Id_{U^{∗}⊗U})
= (k^{′′∗})^{−1}(Ev_{V ⊗U})
= T r_{V ⊗U}.

Remark 2.35. The previous Theorem appears in slightly different form in [K] §II.3. The following Corollary is a well-known identity for the (scalar valued) trace ([B] §II.4.4, [Magnus] §1.10, [K] §II.6).

Corollary 2.36. For A : V → V, B : U → U,

T r_{V ⊗U}(j_{2}(A ⊗ B)) = T r_{V} (A) · T r_{U}(B).

Proof. As in Example 2.25,

T r_{V ⊗U}(j_{2}(A ⊗ B)) = T r_{U}(T r_{V;U,U}(j_{2}(A ⊗ B)))
= T r_{U}(T r_{V} (A) · B)
= T r_{V} (A) · T r_{U}(B).

The result of Corollary 2.36 could also be proved directly using methods similar to the previous proof, and could be stated as the equality

j_{2}^{∗}(T r_{V ⊗U}) = (Hom(Id_{End(V)⊗End(U)}, l) ∘ j)(T r_{V} ⊗ T r_{U}) ∈ (End(V) ⊗ End(U))^{∗},

or

T r_{V ⊗U} ∘ j_{2} = l ∘ [T r_{V} ⊗ T r_{U}].

Corollary 2.37. ([G_{2}] §I.5) For A : V → V, B : U → U,
T r_{Hom(V,U)}(Hom(A, B)) = T r_{V} (A) · T r_{U}(B).

Proof. By Lemma 1.58, Hom(A, B) = k_{V U} ∘ [A^{∗} ⊗ B] ∘ k_{V U}^{−1}, so Lemma 2.6, Corollary 2.36, and Lemma 2.5 apply:

T r_{Hom(V,U)}(Hom(A, B)) = T r_{Hom(V,U)}(k_{V U} ∘ [A^{∗} ⊗ B] ∘ k_{V U}^{−1})
= T r_{V^{∗}⊗U}([A^{∗} ⊗ B])
= T r_{V^{∗}}(A^{∗}) · T r_{U}(B)
= T r_{V} (A) · T r_{U}(B).

Theorem 2.38. For finite-dimensional V and V′, and A : V ⊗ V′ ⊗ U → V ⊗ V′ ⊗ W,

T r_{V ⊗V′;U,W} (A) = T r_{V′;U,W} (T r_{V;V′⊗U,V′⊗W} (A)).

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j′_{1}} & M_{12} \ar[d]^{j′′_{1}} \\ M_{21} \ar[r]^{a_{2}} \ar[d]^{Hom((j′_{2})^{−1},l_{1})} & M_{22} \ar[d]^{Hom((j′′_{2})^{−1},l_{1})} \\ M_{31} \ar[r]^{a_{3}} \ar[u]^{Hom(j′_{2},l_{1}^{−1})} & M_{32} \ar[u]^{Hom(j′′_{2},l_{1}^{−1})}}

the objects are

M_{11} = End(V′)^{∗} ⊗ End(Hom(U, W))
M_{21} = Hom(End(V′)⊗Hom(U, W), 𝕂⊗Hom(U, W))
M_{31} = Hom(Hom(V′ ⊗ U, V′ ⊗ W), Hom(U, W))
M_{12} = End(V ⊗ V′)^{∗} ⊗ End(Hom(U, W))
M_{22} = Hom(End(V ⊗ V′) ⊗ Hom(U, W), 𝕂⊗ Hom(U, W))
M_{32} = Hom(Hom(V ⊗ V′ ⊗ U, V ⊗ V′ ⊗ W), Hom(U, W)),

the horizontal arrows are

a_{1} = [(T r_{V;V′,V′})^{∗} ⊗ Id_{End(Hom(U,W))}]
a_{2} = Hom([T r_{V;V′,V′} ⊗ Id_{Hom(U,W)}], Id_{𝕂⊗Hom(U,W)})
a_{3} = Hom(T r_{V;V′⊗U,V′⊗W} , Id_{Hom(U,W)}),

and the statement of the Theorem is that

a_{3}(T r_{V′;U,W}) = T r_{V ⊗V′;U,W} .

The commutativity of this square,

\xymatrix{End(V)⊗End(V′)⊗Hom(U, W) \ar[r]^{[j′′′_{2}⊗Id_{Hom(U,W)}]} \ar[d]^{[Id_{End(V )}⊗j′_{2}]} & End(V ⊗V′)⊗Hom(U, W) \ar[d]_{j′′_{2}} \\ End(V)⊗Hom(V′⊗U, V′⊗W) \ar[r]^{j_{2}} & Hom(V ⊗V′⊗U, V ⊗V′⊗W)}

is easy to check, and together with that of the diagram:

\xymatrix{End(V)⊗End(V′)⊗Hom(U,W) \ar[r]^{[j′′′_{2}⊗Id_{Hom(U,W)}]} \ar[d]^{[Id_{End(V )}⊗j′_{2}]} & End(V ⊗V′)⊗Hom(U,W) \ar[d]_{[T r_{V ;V ′,V ′} ⊗Id_{Hom(U,W)}]} \\ End(V)⊗Hom(V′⊗U, V′⊗W) \ar[d]^{j_{1}(T r_{V} ⊗Id_{Hom(V ′⊗U,V ′⊗W)})} & End(V′) ⊗ Hom(U, W) \ar[d]^{j′_{2}} \\ 𝕂 ⊗ Hom(V′ ⊗ U, V′ ⊗ W) \ar[r]^{l′_{1}} & Hom(V′ ⊗ U, V′ ⊗ W)}

D ⊗ E ⊗ F ↦ (j′_{2} ∘ [(T r_{V;V′,V′} ∘ j′′′_{2}) ⊗ Id_{Hom(U,W)}])(D⊗E⊗F)
= j′_{2}((T r_{V;V′,V′}(j′′′_{2}(D ⊗ E))) ⊗ F)
= (T r_{V} (D)) · j′_{2}(E ⊗ F)
D ⊗ E ⊗ F ↦ (l′_{1} ∘ (j(T r_{V} ⊗ j′_{2})))(D ⊗ E ⊗ F)
= l′_{1}((T r_{V} (D)) ⊗ (j′_{2}(E ⊗ F)))
= (T r_{V} (D)) · j′_{2}(E ⊗ F),

implies

T r_{V;V′⊗U,V′⊗W} ∘ j′′_{2} = l′_{1} ∘ (j_{1}(T r_{V} ⊗ Id_{Hom(V′⊗U,V′⊗W)})) ∘ j_{2}^{−1} ∘ j′′_{2}
= l′_{1} ∘ (j_{1}(T r_{V} ⊗ Id_{Hom(V′⊗U,V′⊗W)}))
∘[Id_{End(V)} ⊗ j′_{2}] ∘ [(j′′′_{2})^{−1} ⊗ Id_{Hom(U,W)}]
= j′_{2} ∘ [T r_{V;V′,V′} ⊗ Id_{Hom(U,W)}],

which is what is needed to show that the lower square of the first diagram is commutative. Its upper square is commutative by Lemma 1.37, and the distinguished elements in the top row are related by Theorem 2.34:

a_{1}(T r_{V′} ⊗ Id_{Hom(U,W)}) = (T r_{V′} ∘ T r_{V;V′,V′}) ⊗ Id_{Hom(U,W)} = T r_{V⊗V′} ⊗ Id_{Hom(U,W)}.

The Theorem follows:

a_{3}(T r_{V;U,W}) = (a_{3} ∘ Hom((j′_{2})^{−1}, l_{1}) ∘ j′_{1})(T r_{V′} ⊗ Id_{Hom(U,W)})
= (Hom((j′′_{2})^{−1}, l_{1}) ∘ j′′_{1} ∘ a_{1})(T r_{V′} ⊗ Id_{Hom(U,W)})
= (Hom((j′′_{2})^{−1}, l_{1}) ∘ j′′_{1})(T r_{V ⊗V′} ⊗ Id_{Hom(U,W)})
= T r_{V ⊗V′;U,W} .

Remark 2.39. The result of the above Theorem is another “vanishing” property of the generalized trace ([JSV]).

The maps

j_{3} : Hom(V_{1} ⊗ U_{1}, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2} ⊗ U_{2}, V_{2} ⊗ W_{2})
→ Hom(V_{1} ⊗ U_{1} ⊗ V_{2} ⊗ U_{2}, V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2}),
j_{4} : Hom(U_{1}, W_{1})⊗Hom(U_{2}, W_{2}) → Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2})

appear in the following Theorem comparing the trace of a tensor product to the tensor product of traces. There are also some switching maps, as in Theorem 2.34,

s_{1} : V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2} → V_{1} ⊗ V_{2} ⊗ W_{1} ⊗ W_{2}
s_{2} : V_{1} ⊗ V_{2} ⊗ U_{1} ⊗ U_{2} → V_{1} ⊗ U_{1} ⊗ V_{2} ⊗ U_{2}.

Theorem 2.40. For finite-dimensional V_{1}, V_{2}, and maps A : V_{1}⊗U_{1} → V_{1}⊗W_{1} and B : V_{2} ⊗ U_{2} → V_{2} ⊗ W_{2},

T r_{V_{1}⊗V_{2};U_{1}⊗U_{2},W_{1}⊗W_{2}}(s_{1}∘(j_{3}(A⊗B))∘s_{2}) = j_{4}((T r_{V_{1};U_{1},W_{1}}(A))⊗(T r_{V_{2};U_{2},W_{2}}(B))).

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j_{1}} & M_{12} \ar[d]^{j_{5}} & M_{13} \ar[l]_{a_{2}} \ar[d]^{j_{6}} & M_{14} \ar[l]_{[j⊗j]∘s_{3}} \ar[d]^{[j_{1}^{1}⊗j_{1}^{2}]} \\ M_{21} \ar[r]^{a_{3}} \ar[d]^{a_{5}^{−1}} & M_{22} \ar[d]^{a_{6}^{−1}} & M_{23} \ar[l]_{a_{4}} \ar[d]^{a_{7}^{−1}} & M_{24} \ar[l]_{Hom(s_{4},s_{5})∘j_{7}} \ar[d]^{a_{8}^{−1}} \\ M_{31} \ar[u]^{a_{5}} \ar[r]^{a_{9}} & M_{32} \ar[u]^{a_{6}} & M_{33} \ar[u]^{a_{7}} \ar[l]_{a_{10}} & M_{34} \ar[u]^{a_{8}} \ar[l]_{j_{8}}}

the objects are

M_{11} = End(V_{1} ⊗ V_{2})^{∗} ⊗ End(Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{21} = Hom(End(V_{1} ⊗ V_{2}) ⊗ Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}),
𝕂 ⊗ Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{31} = Hom(Hom(V_{1} ⊗ V_{2} ⊗ U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2} ⊗ W_{1} ⊗ W_{2}),
Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{12} = (End(V_{1}) ⊗ End(V_{2}))^{∗} ⊗
Hom(Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}), Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{22} = Hom(End(V_{1}) ⊗ End(V_{2}) ⊗ Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}),
𝕂 ⊗ Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{32} = Hom(Hom(V_{1} ⊗ U_{1}, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2} ⊗ U_{2}, V_{2} ⊗ W_{2}),
Hom(U_{1} ⊗ U_{2}, W_{1} ⊗ W_{2}))
M_{13} = Hom(End(V_{1}) ⊗ End(V_{2}), 𝕂 ⊗ 𝕂) ⊗ End(Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}))
M_{23} = Hom(End(V_{1}) ⊗ End(V_{2}) ⊗ Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}),
𝕂 ⊗ 𝕂 ⊗ Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}))
M_{33} = Hom(Hom(V_{1} ⊗ U_{1}, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2} ⊗ U_{2}, V_{2} ⊗ W_{2}),
Hom(U_{1}, W_{1}) ⊗ Hom(U_{2}, W_{2}))
M_{14} = End(V_{1})^{∗} ⊗ End(Hom(U_{1}, W_{1})) ⊗ End(V_{2})^{∗} ⊗ End(Hom(U_{2}, W_{2}))
M_{24} = Hom(End(V_{1}) ⊗ Hom(U_{1}, W_{1}), 𝕂 ⊗ Hom(U_{1}, W_{1})) ⊗
Hom(End(V_{2}) ⊗ Hom(U_{2}, W_{2}), 𝕂 ⊗ Hom(U_{2}, W_{2}))
M_{34} = Hom(Hom(V_{1} ⊗ U_{1}, V_{1} ⊗ W_{1}), Hom(U_{1}, W_{1})) ⊗
Hom(Hom(V_{2} ⊗ U_{2}, V_{2} ⊗ W_{2}), Hom(U_{2}, W_{2}));

the left, right columns define T r_{V_{1}⊗V_{2};U_{1}⊗U_{2},W_{1}⊗W_{2}} and T r_{V_{1};U_{1},W_{1}} ⊗ T r_{V_{2};U_{2},W_{2}}. The arrows are

a_{1} = [(j′_{2})^{∗} ⊗ Hom(j_{4}, Id_{Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})})]
a_{2} = [Hom(Id_{End(V_{1})⊗End(V_{2})}, l_{𝕂}) ⊗ Hom(Id_{Hom(U_{1},W_{1})⊗Hom(U_{2},W_{2})}, j_{4})]
a_{3} = Hom([j′_{2} ⊗ j_{4}], Id_{𝕂⊗Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})})
a_{4} = Hom(Id_{End(V_{1})⊗End(V_{2})⊗Hom(U_{1},W_{1})⊗Hom(U_{2},W_{2})}, [l_{𝕂} ⊗ j_{4}])
a_{5} = Hom(j_{2}, l_{1}^{−1})
a_{6} = Hom([j_{2}^{1} ⊗ j_{2}^{2}] ∘ s_{4}, l_{1}^{−1})
a_{7} = Hom([j_{2}^{1} ⊗ j_{2}^{2}] ∘ s_{4}, (l ∘ l)^{−1})
a_{8} = [Hom(j_{2}^{1}, (l_{1}^{1})^{−1}) ⊗ Hom(j_{2}^{2}, (l_{1}^{2})^{−1})]
a_{9} = Hom(Hom(s_{2}, s_{1}) ∘ j_{3}, Id_{Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})})
a_{10} = Hom(Id_{Hom(V_{1}⊗U_{1},V_{1}⊗W_{1})⊗Hom(V_{2}⊗U_{2},V_{2}⊗W_{2})}, j_{4}).

The Theorem claims the two maps

T r_{V_{1}⊗V_{2};U_{1}⊗U_{2},W_{1}⊗W_{2}} ∘ Hom(s_{2}, s_{1}) ∘ j_{3} = a_{9}(T r_{V_{1}⊗V_{2};U_{1}⊗U_{2},W_{1}⊗W_{2}}),
j_{4} ∘ (j_{8}(T r_{V_{1};U_{1},W_{1}} ⊗ T r_{V_{2};U_{2},W_{2}})) = (a_{10} ∘ j_{8})(T r_{V_{1};U_{1},W_{1}} ⊗ T r_{V_{2};U_{2},W_{2}})

are equal. The diagram is commutative— all six squares are easy to check, for example, the upper left and upper middle follow from Lemma 1.37, and each of the remaining four involves two arrows with switching maps. The equality along the top row,

a_{1} : T r_{V_{1}⊗V_{2}} ⊗ Id_{Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})} ↦ (T r_{V_{1}⊗V_{2}} ∘ j′_{2}) ⊗ j_{4},
a_{2} ∘ [j ⊗ j] ∘ s_{3} : T r_{V_{1}} ⊗ Id_{Hom(U_{1},W_{1})} ⊗ T r_{V_{2}} ⊗ Id_{Hom(U_{2},W_{2})}
↦ (l_{𝕂} ∘ [T r_{V_{1}} ⊗ T r_{V_{2}}]) ⊗ j_{4},

follows from Corollary 2.36. This key step, together with the commutativity of the diagram, proves the Theorem:

a_{9} : (T r_{V_{1}⊗V_{2};U_{1}⊗U_{2},W_{1}⊗W_{2}})
↦ (a_{9} ∘ a_{5}^{−1} ∘ j_{1})(T r_{V_{1}⊗V_{2}} ⊗ Id_{Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})})
= (a_{6}^{−1} ∘ j_{5} ∘ a_{1})(T r_{V_{1}⊗V_{2}} ⊗ Id_{Hom(U_{1}⊗U_{2},W_{1}⊗W_{2})})
= (a_{6}^{−1} ∘ j_{5} ∘ a_{2} ∘ [j ⊗ j] ∘ s_{3})(T r_{V_{1}} ⊗ Id_{Hom(U_{1},W_{1})} ⊗ T r_{V_{2}} ⊗ Id_{Hom(U_{2},W_{2})})
= (a_{10} ∘ j_{8} ∘ a_{8}^{−1} ∘ [j_{1}^{1} ⊗ j_{1}^{2}])(T r_{V_{1}} ⊗ Id_{Hom(U_{1},W_{1})} ⊗ T r_{V_{2}} ⊗ Id_{Hom(U_{2},W_{2})})
= (a_{10} ∘ j_{8})(T r_{V_{1};U_{1},W_{1}} ⊗ T r_{V_{2};U_{2},W_{2}}).

Remark 2.41. The maps j_{4}, from the previous Theorem, and

j_{0} : Hom(V ⊗ U_{1}, V ⊗ W_{1}) ⊗ Hom(U_{2}, W_{2}) → Hom(V ⊗ U_{1} ⊗ U_{2}, V ⊗ W_{1} ⊗ W_{2})

appear in the following Corollary about the compatibility of the trace and the tensor product, related to a “superposing” identity of [JSV].

Corollary 2.42. For A : V ⊗ U_{1} → V ⊗ W_{1} and B : U_{2} → W_{2},

T r_{V;U_{1}⊗U_{2},W_{1}⊗W_{2}}(j_{0}(A ⊗ B)) = j_{4}((T r_{V;U_{1},W_{1}}(A)) ⊗ B).

Proof. It can be checked that the following diagram is commutative.

\xymatrix{V ⊗ U_{1} ⊗ 𝕂 ⊗ U_{2} \ar[r]^{j_{3}(A⊗(l_{W_{2}}^{−1} ∘B∘l_{U_{2}} ))} & V ⊗ W_{1} ⊗ 𝕂 ⊗ W_{2} \ar[d]^{s_{1}} \\ V ⊗ 𝕂 ⊗ U_{1} ⊗ U_{2} \ar[u]^{s_{2}} \ar[d]^{[l_{V} ⊗Id_{U_{1}⊗U_{2}} ]} & V ⊗ 𝕂 ⊗ W_{1} ⊗ W_{2} \ar[d]^{[l_{V} ⊗Id_{W_{1}⊗W_{2}} ]} \\ V ⊗ U_{1} ⊗ U_{2} \ar[r]^{j_{0}(A⊗B)} & V ⊗ W_{1} ⊗ W_{2}}

Theorem 2.29, the diagram, the previous Theorem, and finally Theorem 2.26 apply:

LHS = T r_{V;U_{1}⊗U_{2},W_{1}⊗W_{2}}((j_{0}(A ⊗ B)) ∘ [l_{V} ⊗ Id_{U_{1}⊗U_{2}}] ∘ [l_{V}^{−1} ⊗ Id_{U_{1}⊗U_{2}}])
= T r_{V ⊗𝕂;U_{1}⊗U_{2},W_{1}⊗W_{2}}([l_{V}^{−1} ⊗ Id_{W_{1}⊗W_{2}}] ∘ (j_{0}(A ⊗ B)) ∘ [l_{V} ⊗ Id_{U_{1}⊗U_{2}}])
= T r_{V ⊗𝕂;U_{1}⊗U_{2},W_{1}⊗W_{2}}(s_{1} ∘ (j_{3}(A ⊗ (l_{W_{2}}^{−1} ∘ B ∘ l_{U_{2}}))) ∘ s_{2})
= j_{4}((T r_{V;U_{1},W_{1}}(A)) ⊗ (T r_{𝕂;U_{2},W_{2}}(Hom(l_{U_{2}}, l_{W_{2}}^{−1})(B))))
= j_{4}((T r_{V;U_{1},W_{1}}(A)) ⊗ B).

Notation 2.43. Denote the composite maps

j̃_{U} = Hom(Id_{V ⊗U}, l) ∘ j : V^{∗} ⊗ U^{∗} → (V ⊗ U)^{∗}
j̃_{W} = Hom(Id_{V ⊗W} , l) ∘ j : V^{∗} ⊗ W^{∗} → (V ⊗ W)^{∗},

where l is multiplication 𝕂 ⊗ 𝕂 → 𝕂.

If V is finite-dimensional then these j̃ maps are invertible. They appear in the next Theorem, relating the trace of the transpose to the transpose of the trace, and already appeared in Lemma 2.32. Such composites appear again in the next Chapter, but the tilde notation will only be used when abbreviating is more useful than not.

Theorem 2.44. For finite-dimensional V, H : V ⊗ U → V ⊗ W, and t_{UV}, t_{V ⊗U,V ⊗W} as in Notation 1.9,

T r_{V^{∗};W^{∗},U^{∗}}(j̃_{U}^{−1} ∘ (t_{V ⊗U,V ⊗W} (H)) ∘ j̃_{W}) = t_{UW} (T r_{V;U,W} (H)).

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j_{1}} & M_{12} \ar[d]^{j′′_{1}} & M_{13} \ar[l]_{a_{2}} \ar[d]^{j′_{1}} \\ M_{21} \ar[r]^{a_{3}} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & M_{22} \ar[d]^{Hom(j_{2}^{−1},l′_{1})} & M_{23} \ar[l]_{a_{4}} \ar[d]^{Hom((j′_{2})^{−1},l′_{1})} \\ M_{31} \ar[u]^{Hom(j_{2},l_{1}^{−1})} \ar[r]^{a_{5}} & M_{32} \ar[u]^{Hom(j_{2},(l′_{1})^{−1})} & M_{33} \ar[u]^{Hom(j′_{2},(l′_{1})^{−1})} \ar[l]_{a_{6}}}

2.2. THE GENERALIZED TRACE 67

the objects are

M_{11} = End(V)^{∗} ⊗ End(Hom(U, W))
M_{21} = Hom(End(V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(U, W))
M_{31} = Hom(Hom(V ⊗ U, V ⊗ W), Hom(U, W))
M_{12} = End(V)^{∗} ⊗ Hom(Hom(U, W), Hom(W^{∗}, U^{∗}))
M_{22} = Hom(End(V) ⊗ Hom(U, W), 𝕂 ⊗ Hom(W^{∗}, U^{∗}))
M_{32} = Hom(Hom(V ⊗ U, V ⊗ W), Hom(W^{∗}, U^{∗}))
M_{13} = End(V^{∗})^{∗} ⊗ End(Hom(W^{∗}, U^{∗}))
M_{23} = Hom(End(V^{∗}) ⊗ Hom(W^{∗}, U^{∗}), 𝕂 ⊗ Hom(W^{∗}, U^{∗}))
M_{33} = Hom(Hom(V^{∗} ⊗ W^{∗}, V^{∗} ⊗ U^{∗}), Hom(W^{∗}, U^{∗})),

the arrows are

a_{1} = [Id_{End(V)^{∗}} ⊗ Hom(Id_{Hom(U,W)}, t_{UW})]
a_{2} = [t_{V V}^{∗} ⊗ Hom(t_{UW} , Id_{Hom(W^{∗},U^{∗})})]
a_{3} = Hom(Id_{End(V)⊗Hom(U,W)}, [Id_{𝕂} ⊗ t_{UW}])
a_{4} = Hom([t_{V V} ⊗ t_{UW}], Id_{𝕂⊗Hom(W^{∗},U^{∗})})
a_{5} = Hom(Id_{Hom(V ⊗U,V ⊗W)}, t_{UW})
a_{6} = Hom(Hom(j̃_{W} , j̃_{U}^{−1}) ∘ t_{V ⊗U,V ⊗W} , Id_{Hom(W^{∗},U^{∗})}),

and the two quantities in the statement of the Theorem are

t_{UW} (T r_{V;U,W} (H)) = (a_{5}(T r_{V;U,W}))(H)
T r_{V^{∗};W^{∗},U^{∗}}(Hom(j̃_{W} , j̃_{U}^{−1})(t_{V ⊗U,V ⊗W} (H))) = (a_{6}(T r_{V^{∗};W^{∗},U^{∗}}))(H).

The diagram is commutative, for example, the lower right square:

E ⊗ F ↦ (Hom(Id_{V^{∗}⊗W^{∗}}, j̃_{U}) ∘ j′_{2} ∘ [t_{V V} ⊗ t_{UW}])(E ⊗ F)
= j̃_{U} ∘ (j′_{2}((t_{V V} (E)) ⊗ (t_{UW} (F)))) :
ϕ ⊗ ξ ↦ j̃_{U}((E^{∗}(ϕ)) ⊗ (F^{∗}(ξ))) :
v ⊗ u ↦ ϕ(E(v)) · ξ(F (u)),
E ⊗ F ↦ (Hom(j̃_{W} , Id_{V^{∗}⊗U^{∗}}) ∘ t_{V ⊗U,V ⊗W} ∘ j_{2})(E ⊗ F)
= (t_{V ⊗U,V ⊗W} (j_{2}(E ⊗ F))) ∘ j̃_{W} :
ϕ ⊗ ξ ↦ (j̃_{W} (ϕ ⊗ ξ)) ∘ (j_{2}(E ⊗ F)) :
v ⊗ u ↦ ϕ(E(v)) · ξ(F (u)).

Lemma 2.5 implies the equality of the outputs of the distinguished elements in the top row:

a_{1}(T r_{V} ⊗ Id_{Hom(U,W)}) = T r_{V} ⊗ t_{UW}
a_{2}(T r_{V^{∗}} ⊗ Id_{Hom(W^{∗},U^{∗})}) = (t_{V V}^{∗} (T r_{V^{∗}})) ⊗ t_{UW}
= T r_{V} ⊗ t_{UW} ,

68 2. A SURVEY OF TRACE ELEMENTS

and the Theorem follows from the commutativity of the diagram:

a_{5}(T r_{V;U,W}) = (a_{5} ∘ Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom(j_{2}^{−1}, l′_{1}) ∘ j′′_{1} ∘ a_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom(j_{2}^{−1}, l′_{1}) ∘ j′′_{1} ∘ a_{2})(T r_{V^{∗}} ⊗ Id_{Hom(W^{∗},U^{∗})})
= (a_{6} ∘ Hom((j′_{2})^{−1}, l′_{1}) ∘ j′_{1})(T r_{V^{∗}} ⊗ Id_{Hom(W^{∗},U^{∗})})
= a_{6}(T r_{V^{∗};W^{∗},U^{∗}}).

Exercise 2.45. Given a direct sum V = V_{1} ⊕ V_{2}, and A : V_{1} ⊗ U → V_{1} ⊗ W, and B : V_{2} ⊗ U → V_{2} ⊗ W, define A ⊕ B : V ⊗ U → V ⊗ W using the projections and inclusions from Example 1.77:

A ⊕ B = [Q_{1} ⊗ Id_{W}] ∘ A ∘ [P_{1} ⊗ Id_{U}] + [Q_{2} ⊗ Id_{W}] ∘ B ∘ [P_{2} ⊗ Id_{U}]. If V is finite-dimensional, then

T r_{V;U,W} (A ⊕ B) = T r_{V_{1};U,W} (A) + T r_{V_{2};U,W} (B). Hint. The proof proceeds exactly as in Proposition 2.12, using Theorem 2.29. Exercise 2.46. For V = V_{1} ⊕ V_{2} as above, and K : V ⊗ U → V ⊗ W,

T r_{V;U,W} (K) = T r_{V_{1};U,W} ([P_{1} ⊗ Id_{W}] ∘ K ∘ [Q_{1} ⊗ Id_{U}])
+T r_{V_{2};U,W} ([P_{2} ⊗ Id_{W}] ∘ K ∘ [Q_{2} ⊗ Id_{U}]).

Hint. Using Theorem 2.29 and Lemma 1.36,

T r_{V_{i};U,W} ([P_{i} ⊗ Id_{W}] ∘ K ∘ [Q_{i} ⊗ Id_{U}]) = T r_{V;U,W} ([(Q_{i} ∘ P_{i}) ⊗ Id_{W}] ∘ K). The proof proceeds exactly as in Proposition 2.13.

Proposition 2.47. For finite-dimensional V_{1}, V_{2}, maps A : V_{1}⊗U_{1} → V_{2}⊗W_{2}, B : V_{2} ⊗ U_{2} → V_{1} ⊗ W_{1}, and switching maps as in the following diagrams,

\xymatrix{V_{1} ⊗ U_{2} ⊗ U_{1} \ar[d]^{s_{1}} & V_{2} ⊗ U_{2} ⊗ U_{1} \ar[d]^{[B⊗Id_{U_{1}} ]} \\ V_{1} ⊗ U_{1} ⊗ U_{2} \ar[d]^{[A⊗Id_{U_{2}} ]} & V_{1} ⊗ W_{1} ⊗ U_{1} \ar[d]^{s_{3}} \\ V_{2} ⊗ W_{2} ⊗ U_{2} \ar[d]^{s_{2}} & V_{1} ⊗ U_{1} ⊗ W_{1} \ar[d]^{[A⊗Id_{W_{1}} ]} \\ V_{2} ⊗ U_{2} ⊗ W_{2} \ar[d]^{[B⊗Id_{W_{2}} ]} & V_{2} ⊗ W_{2} ⊗ W_{1} \ar[d]^{s_{4}} \\ V_{1} ⊗ W_{1} ⊗ W_{2} & V_{2} ⊗ W_{1} ⊗ W_{2}}

the traces of the composites are equal:

T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}([B ⊗ Id_{W_{2}}] ∘ s_{2} ∘ [A ⊗ Id_{U_{2}}] ∘ s_{1})

= T r_{V_{2};U_{2}⊗U_{1},W_{1}⊗W_{2}}(s_{4} ∘ [A ⊗ Id_{W_{1}}] ∘ s_{3} ∘ [B ⊗ Id_{U_{1}}]).

Proof. The canonical map

j : Hom(V_{2}, V_{1}) ⊗ Hom(U_{2}, W_{1}) → Hom(V_{2} ⊗ U_{2}, V_{1} ⊗ W_{1})

is invertible by the finite-dimensionality hypothesis and Claim 1.34. Also consider a map

Q_{1}^{1} : 𝕂 → End(W_{2}) : 1 ↦ Id_{W_{2}},

as in (2.4) (and again not necessarily from some direct sum as in Example 2.9), and analogously

Q̃_{1}^{1} : 𝕂 → End(U_{1}) : 1 ↦ Id_{U_{2}}.

The claim of the Proposition is that the lower block of this diagram is commutative, starting with B ⊗ 1 in the upper right.

\xymatrix{Hom(V_{2}, V_{1}) ⊗ Hom(U_{2}, W_{1}) ⊗ 𝕂 \ar[r]^{[j⊗Id_{𝕂}]} \ar[d]_{[j⊗Q̃_{1}^{1}]} & Hom(V_{2} ⊗ U_{2}, V_{1} ⊗ W_{1}) ⊗ 𝕂 \ar@{-->}[dl]_{[Id_{Hom(V_{2}⊗U_{2},V_{1}⊗W_{1})}⊗Q̃_{1}^{1}]} \ar[d]_{[Id_{Hom(V_{2}⊗U_{2},V_{1}⊗W_{1})}⊗Q_{1}^{1}]} \\ Hom(V_{2} ⊗ U_{2}, V_{1} ⊗ W_{1}) ⊗ End(U_{1}) \ar[d]_{j′′} & Hom(V_{2} ⊗ U_{2}, V_{1} ⊗ W_{1}) ⊗ End(W_{2}) \ar[d]^{j′} \\ Hom(V_{2} ⊗ U_{2} ⊗ U_{1}, V_{1} ⊗ W_{1} ⊗ U_{1}) \ar[d]^{Hom(Id_{V_{2}⊗U_{2}⊗U_{1}} ,s_{4}⊗[A⊗Id_{W_{1}} ]∘s_{3})} & Hom(V_{2} ⊗ U_{2} ⊗ W_{2}, V_{1} ⊗ W_{1} ⊗ W_{2}) \ar[d]_{Hom(s_{2}⊗[A⊗Id_{U_{2}} ]∘s_{1},Id_{V_{1}⊗W_{1}⊗W_{2}} )} \\ Hom(V_{2} ⊗ U_{2} ⊗ U_{1}, V_{2} ⊗ W_{1} ⊗ W_{2}) \ar[d]^{T r_{V_{2};U_{2}⊗U_{1},W_{1}⊗W_{2}}} & Hom(V_{1} ⊗ U_{2} ⊗ U_{1}, V_{1} ⊗ W_{1} ⊗ W_{2}) \ar@{-->}[dl]^{T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}} \\ Hom(U_{2} ⊗ U_{1}, W_{1} ⊗ W_{2})}

The upper triangle of the above diagram is commutative, so to prove the claim it is enough, by the invertibility of the upper arrow [j ⊗ Id_{𝕂}], to check the outside of the diagram, starting with B_{1} ⊗ B_{2} ⊗ 1 in the upper left, for B_{1} : V_{2} → V_{1}, B_{2} : U_{2} → W_{1}, so that the downward composite on the left side is equal to the composite path going clockwise around the right.

The following easily checked diagram shows how the B_{2} factor commutes with A.

\xymatrix{V_{1} ⊗ U_{2} ⊗ U_{1} \ar[d]^{s_{1}} \ar[r]^{[Id_{V_{1}} ⊗[B_{2}⊗Id_{U_{1}} ]]} & V_{1} ⊗ W_{1} ⊗ U_{1} \ar[d]^{s_{3}} \\ V_{1} ⊗ U_{1} ⊗ U_{2} \ar[d]^{[A⊗Id_{U_{2}} ]} \ar[r]^{[Id_{V_{1}⊗U_{1}} ⊗B_{2}]} & V_{1} ⊗ U_{1} ⊗ W_{1} \ar[d]^{[A⊗Id_{W_{1}} ]} \\ V_{2} ⊗ W_{2} ⊗ U_{2} \ar[d]^{s_{2}} \ar[r]^{[Id_{V_{2}⊗W_{2}} ⊗B_{2}]} & V_{2} ⊗ W_{2} ⊗ W_{1} \ar[d]^{s_{4}} \\ V_{2} ⊗ U_{2} ⊗ W_{2} \ar[r]^{[Id_{V_{2}} ⊗[B_{2}⊗Id_{W_{2}} ]]} & V_{2} ⊗ W_{1} ⊗ W_{2}}

Using Theorem 2.29,

T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}([[B_{1} ⊗ B_{2}] ⊗ Id_{W_{2}}] ∘ s_{2} ∘ [A ⊗ Id_{U_{2}}] ∘ s_{1})

= T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}([B_{1}⊗ Id_{W_{1}⊗W_{2}}]∘ [Id_{V_{2}} ⊗ [B_{2} ⊗ Id_{W_{2}}]] ∘ s_{2}∘ [A ⊗ Id_{U_{2}}] ∘ s_{1})
= T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}([B_{1}⊗ Id_{W_{1}⊗W_{2}}] ∘ s_{4}∘ [A ⊗ Id_{W_{1}}] ∘ s_{3}∘ [Id_{V_{1}} ⊗ [B_{2} ⊗ Id_{U_{1}}]])
= T r_{V_{2};U_{2}⊗U_{1},W_{1}⊗W_{2}}(s_{4}∘ [A ⊗ Id_{W_{1}}] ∘ s_{3}∘ [Id_{V_{1}} ⊗ [B_{2}⊗ Id_{U_{1}}]] ∘ [B_{1}⊗ Id_{U_{2}⊗U_{1}}])
= T r_{V_{2};U_{2}⊗U_{1},W_{1}⊗W_{2}}(s_{4} ∘ [A ⊗ Id_{W_{1}}] ∘ s_{3} ∘ [[B_{1} ⊗ B_{2}] ⊗ Id_{U_{1}}]).

Exercise 2.48. Using various switching maps, Proposition 2.47 can be used to prove related identities. For example, given maps:

A′ : U_{1} ⊗ V_{1} → V_{2} ⊗ W_{2}
B′ : U_{2} ⊗ V_{2} → V_{1} ⊗ W_{1}
s_{5} : V_{1} ⊗ U_{2} ⊗ U_{1} → U_{2} ⊗ U_{1} ⊗ V_{1}
s_{6} : V_{2} ⊗ U_{2} ⊗ U_{1} → U_{1} ⊗ U_{2} ⊗ V_{2},

the following identity can be proved as a consequence of Proposition 2.47:

T r_{V_{1};U_{2}⊗U_{1},W_{1}⊗W_{2}}([B′ ⊗ Id_{W_{2}}] ∘ [Id_{U_{2}} ⊗ A′] ∘ s_{5})
(2.7)  = T r_{V_{2};U_{2}⊗U_{1},W_{1}⊗W_{2}}(s_{4} ∘ [A′ ⊗ Id_{W_{1}}] ∘ [Id_{U_{1}} ⊗ B′] ∘ s_{6}).

Hint. Let A = A′ ∘ s and B = B′ ∘ s for appropriate switching maps s.

Remark 2.49. Equation (2.7) is related to [PS] Proposition 2.7, on the “cyclicity” of the generalized trace.

2.3. Vector valued trace

In analogy with Definition 2.24, but with no space U, the “vector valued” or “W -valued” trace of a map V → V ⊗W should be an element of W. The results on the generalized trace have analogues in this case, but the construction uses different canonical maps.

2.3.1. Defining the vector valued trace.

The following Definition 2.50 applies to an arbitrary vector space W and a finite-dimensional space V to define the W -valued trace T r_{V;W} : Hom(V, V ⊗W) → W, in terms of the previously defined (scalar) trace T r_{V}, and canonical maps

n : End(V) ⊗ W → Hom(V, V ⊗ W)
j′_{1} : End(V)^{∗} ⊗ End(W) → Hom(End(V) ⊗ W, 𝕂 ⊗ W),

where j′_{1} is another canonical j map in analogy with j_{1} from Definition 2.24, and n is invertible by Lemma 1.43.

Definition 2.50. T r_{V;W} = (Hom(n^{−1}, l_{W}) ∘ j′_{1})(T r_{V} ⊗ Id_{W}).

Example 2.51. A map of the form n(A ⊗ w) : V → V ⊗ W, for A : V → V and w ∈ W, has trace

T r_{V;W} (n(A ⊗ w)) = l_{W} ((j′_{1}(T r_{V} ⊗ Id_{W}))(A ⊗ w)) = T r_{V} (A) · w.

A map q : Hom(V, Hom(U, V ⊗ W)) → Hom(V ⊗ U, V ⊗ W) from Definition 1.44 and a map n′ : V ⊗Hom(U, W) → Hom(U, V ⊗W) as in Notation 1.40 are used in the following Theorem relating Definition 2.50 to Definition 2.24. Consider the vector valued trace, in the case where W is replaced by the vector space Hom(U, W).

Theorem 2.52. For finite-dimensional V, and a map K : V → V ⊗Hom(U, W),

T r_{V;Hom(U,W)}(K) = T r_{V;U,W} (q(n′ ∘ K)).

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{Hom(n^{−1},l_{1})} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & M_{12} \ar[l]^{Hom(n,l_{1}^{−1})} \\ M_{21} \ar[r]^{Hom(q,Id_{Hom(U,W)})} \ar[u]^{Hom(j_{2},l_{1}^{−1})} & M_{22} \ar[u]_{Hom(Hom(Id_{V} ,n′),Id_{Hom(U,W)})}}

the objects are

M_{11} = Hom(End(V)⊗Hom(U,W), 𝕂⊗Hom(U,W))
M_{21} = Hom(Hom(V ⊗U, V ⊗W), Hom(U,W))
M_{12} = Hom(Hom(V, V ⊗ Hom(U, W)), Hom(U, W))
M_{22} = Hom(Hom(V, Hom(U, V ⊗ W)), Hom(U, W)).

The square is commutative since

E ⊗ F ↦ (q ∘ Hom(Id_{V} , n′) ∘ n)(E ⊗ F) = q(n′ ∘ (n(E ⊗ F))) :
v ⊗ u ↦ (n′((n(E ⊗ F))(v)))(u)
= (n′((E(v)) ⊗ F))(u)
= (E(v)) ⊗ (F (u))
= (j_{2}(E ⊗ F))(v ⊗ u).

Definitions 2.24 and 2.50 are related in this case by l_{W} = l_{1} and j′_{1} = j_{1}. The Theorem follows:

T r_{V;Hom(U,W)} = (Hom(n^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= (Hom(j_{2}^{−1} ∘ q ∘ Hom(Id_{V} , n′), l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(U,W)})
= Hom(q ∘ Hom(Id_{V} , n′), Id_{Hom(U,W)})(T r_{V;U,W}).

72 2. A SURVEY OF TRACE ELEMENTS

Definition 2.50 is related to the original definition of trace when W = 𝕂. Theorem 2.53. For H : V → V ⊗ 𝕂, T r_{V} (l_{V} ∘ H) = T r_{V;𝕂}(H). Proof. Let l_{2} : End(V)^{∗} ⊗ 𝕂 → End(V)^{∗} be another scalar multiplication

map. The following diagram is commutative.

\xymatrix{End(V)^{∗} ⊗ 𝕂^{∗} \ar[d]^{j′_{1}} & End(V)^{∗} ⊗ 𝕂 \ar[dd]^{l_{2}} \ar[l]^{[Id_{End(V )^{∗}} ⊗m]} \\ Hom(End(V) ⊗ 𝕂, 𝕂 ⊗ 𝕂) \ar[d]^{Hom(n^{−1},l_{𝕂})} \\ Hom(V, V ⊗ 𝕂)^{∗} \ar[u]^{Hom(n,l_{𝕂}^{−1})} & End(V)^{∗} \ar[l]^{Hom(Id_{V} ,l_{V} )^{∗}}}

For λ, μ ∈ 𝕂, Φ ∈ End(V)^{∗}, A ∈ End(V),

Φ ⊗ λ ↦ (j′_{1} ∘ [Id_{End(V)^{∗}} ⊗ m])(Φ ⊗ λ) = j′_{1}(Φ ⊗ (m(λ))) :
A ⊗ μ ↦ (Φ(A)) ⊗ (μ · λ),
Φ ⊗ λ ↦ (Hom(n, l_{𝕂}^{−1}) ∘ Hom(Id_{V} , l_{V})^{∗} ∘ l_{2})(Φ ⊗ λ)
= l_{𝕂}^{−1}∘ ((λ·Φ) ∘ Hom(Id_{V} , l_{V}))∘n :
A ⊗ μ ↦ l_{𝕂}^{−1}((λ · Φ)(l_{V} ∘ (n(A ⊗ μ)))) = 1 ⊗ (λ · Φ(μ · A)),

since (l_{V} ∘ (n(A ⊗ μ))) : v ↦ l_{V} ((A(v)) ⊗ μ) = (μ · A)(v). The Theorem follows from [Id_{End(V)^{∗}} ⊗ m](T r_{V} ⊗ 1) = T r_{V} ⊗ Id_{𝕂}:

Hom(Id_{V} , l_{V})^{∗}(T r_{V}) = (Hom(Id_{V} , l_{V})^{∗} ∘ l_{2})(T r_{V} ⊗ 1)
= (Hom(n^{−1}, l_{𝕂}) ∘ j′_{1} ∘ [Id_{End(V)^{∗}} ⊗ m])(T r_{V} ⊗ 1)
= T r_{V;𝕂}.

Definition 2.24 and Definition 2.50 are related in the case U = 𝕂: Theorem 2.54. For H : V → V ⊗ W, T r_{V;𝕂,W} (H ∘ l_{V}) : 1 ↦ T r_{V;W} (H). Proof. The following diagram is commutative:

\xymatrix{End(V)^{∗}⊗End(Hom(𝕂,W)) \ar[d]^{j_{1}} \ar[r]^{a_{1}} & End(V)^{∗}⊗End(W) \ar[d]^{j′_{1}} \\ Hom(End(V)⊗Hom(𝕂,W), 𝕂⊗Hom(𝕂,W)) \ar[r]^{a_{2}} \ar[d]^{Hom(j_{2}^{−1},l_{1})} & Hom(End(V)⊗W, 𝕂⊗W) \ar[d]^{Hom(n^{−1},l_{W})} \\ Hom(Hom(V ⊗𝕂, V ⊗W), Hom(𝕂,W)) \ar[r]^{a_{3}} \ar[u]^{Hom(j_{2},l_{1}^{−1})} & Hom(Hom(V, V ⊗W),W) \ar[u]^{Hom(n,l_{W}^{−1})}}

where the horizontal arrows are

a_{1} = [Id_{End(V)^{∗}} ⊗ Hom(m, m^{−1})]
a_{2} = Hom([Id_{End(V)} ⊗ m], [Id_{𝕂} ⊗ m^{−1}])
a_{3} = Hom(Hom(l_{V} , Id_{V ⊗W}), m^{−1}).

The statement of the Theorem becomes

a_{3}(T r_{V;𝕂,W}) = T r_{V;W} .

The top square is commutative by Lemma 1.37. The lower square is commutative by Lemma 1.6, Lemma 1.38, and Lemma 1.42. The Theorem follows from a_{1}(T r_{V} ⊗ Id_{Hom(𝕂,W)}) = T r_{V} ⊗ Id_{W}:

a_{3}(T r_{V;𝕂,W}) = (a_{3} ∘ Hom(j_{2}^{−1}, l_{1}) ∘ j_{1})(T r_{V} ⊗ Id_{Hom(𝕂,W)})
= (Hom(n^{−1}, l_{W}) ∘ j′_{1} ∘ a_{1})(T r_{V} ⊗ Id_{Hom(𝕂,W)})
= (Hom(n^{−1}, l_{W}) ∘ j′_{1})(T r_{V} ⊗ Id_{W})
= T r_{V;W} .

Exercise 2.55. The result of Theorem 2.27,

T r_{V} (H) = (T r_{V;𝕂,𝕂}(l_{V}^{−1} ∘ H ∘ l_{V}))(1),

can be given a different proof using the vector valued trace as an intermediate step.

Hint. By Theorem 2.53 and Theorem 2.54,

T r_{V} = Hom(Id_{V} , l_{V}^{−1})^{∗}(Hom(Hom(l_{V} , Id_{V ⊗𝕂}), m^{−1})(T r_{V;𝕂,𝕂})).

Corollary 2.56. For H : 𝕂 → 𝕂 ⊗ W, T r_{𝕂;W} (H) = l_{W} (H(1)).

Proof. Theorem 2.54 applies, with l_{𝕂} : 𝕂 ⊗ 𝕂 → 𝕂:

T r_{𝕂;W} (H) = (T r_{𝕂;𝕂,W} (H ∘ l_{𝕂}))(1).

By Theorem 2.26, this quantity is equal to

(Hom(l_{𝕂}^{−1}, l_{W})(H ∘ l_{𝕂}))(1) = l_{W} (H(1)).

2.3.2. Properties of the vector valued trace.

The following results on the W -valued trace are corollaries of results from Section 2.2. In most cases, Theorem 2.54 applies, leading to a straightforward calculation.

Corollary 2.57. For A : V → V′, B : V′ → V ⊗ W,

T r_{V;W} (B ∘ A) = T r_{V′;W} ((j_{W} (A ⊗ Id_{W})) ∘ B).

Proof. Theorem 2.29 and Theorem 2.54 apply, using the map l_{V′} : V′ ⊗ 𝕂 → V′, and the equation A ∘ l_{V} = l_{V′} ∘ (j_{𝕂}(A ⊗ Id_{𝕂})), a version of Lemma 1.38.

T r_{V;W} (B ∘ A) = (T r_{V′;𝕂,W} (B ∘ A ∘ l_{V}))(1)
= (T r_{V;𝕂,W} (B ∘ l_{V′} ∘ (j_{𝕂}(A ⊗ Id_{𝕂}))))(1)
= (T r_{V′;𝕂,W} ((j_{W} (A ⊗ Id_{W})) ∘ B ∘ l_{V′}))(1)
= T r_{V′,W} ((j_{W} (A ⊗ Id_{W})) ∘ B).

Corollary 2.58. For A : V → V ⊗ W, B : W → W′,

T r_{V;W′}([Id_{V} ⊗ B] ∘ A) = B(T r_{V;W} (A)).

Proof. By Theorem 2.30 and Theorem 2.54,

T r_{V;W′}([Id_{V} ⊗ B] ∘ A) = (T r_{V;𝕂,W′}([Id_{V} ⊗ B] ∘ A ∘ l_{V}))(1)
= (B ∘ (T r_{V;𝕂,W} (A ∘ l_{V})))(1) = B(T r_{V;W} (A)).

Corollary 2.59. If V and V′ are finite-dimensional then for any maps A : V → V′, B : W → W′, the following diagram is commutative.

\xymatrix{Hom(V, V ⊗ W) \ar[r]^{T r_{V ;W}} & W \ar[dd]^{B} \\ Hom(V′, V ⊗ W) \ar[u]^{Hom(A,Id_{V ⊗W})} \ar[d]_{Hom(Id_{V ′} ,[A⊗B])} \\ Hom(V′, V′ ⊗ W′) \ar[r]^{T r_{V ′;W′}} & W′}

Proof. This follows from Corollary 2.57 and Corollary 2.58.

Lemma 2.60. For a direct sum W = W_{1} ⊕ W_{2} with operators P_{i}, Q_{i}, there is also a direct sum Hom(V, V ⊗ W) = Hom(V, V ⊗ W_{1}) ⊕ Hom(V, V ⊗ W_{2}), with projections Hom(Id_{V} , [Id_{V} ⊗ P_{i}]) and inclusions Hom(Id_{V} , [Id_{V} ⊗ Q_{i}]). If V is finite-dimensional, then the map T r_{V;W} : Hom(V, V ⊗ W) → W respects the direct sums, and each induced map is equal to a W_{i}-valued trace:

T r_{V;W_{i}} = P_{i} ∘ T r_{V;W} ∘ Hom(Id_{V} , [Id_{V} ⊗ Q_{i}]) : Hom(V, V ⊗ W_{i}) → W_{i}.

Proof. The projections and inclusions are as in Example 1.77 and Example 1.78. The claims follow from Definition 1.84 and Corollary 2.58.

Corollary 2.61. For A : V ⊗ V′ → V ⊗ V′ ⊗ W,

T r_{V ⊗V′;W} (A) = T r_{V′;W} (T r_{V;V′,V′⊗W} (A)).

Proof. Using Theorem 2.30, Theorem 2.38, and Theorem 2.54, and the scalar multiplication identity l_{V ⊗V′} = [Id_{V} ⊗ l_{V′}] : V ⊗ V′ ⊗ 𝕂 → V ⊗ V′,

T r_{V ⊗V′;W} (A) = (T r_{V ⊗V′;𝕂,W} (A ∘ l_{V ⊗V′}))(1)
= (T r_{V ⊗V′;𝕂,W} (A ∘ [Id_{V} ⊗ l_{V′}]))(1)
= (T r_{V′;𝕂,W} (T r_{V;V′⊗𝕂,V′⊗W} (A ∘ [Id_{V} ⊗ l_{V′}])))(1)
= (T r_{V′;𝕂,W} ((T r_{V;V′,V′⊗W} (A)) ∘ l_{V′}))(1)
= T r_{V′;W} (T r_{V;V′,V′⊗W} (A)).

2.3. VECTOR VALUED TRACE 75

Exercise 2.62. Denote by n′′ the map

n′′ : W ⊗ Hom(U, W′) → Hom(U, W ⊗ W′)
: w ⊗ E ↦ (u ↦ w ⊗ (E(u))).

Then, for A : V → V ⊗ W, B : U → W′,

T r_{V;U,W ⊗W′}(j′_{0}(A ⊗ B)) = n′′((T r_{V;W} (A)) ⊗ B).

Hint. By Theorem 2.30, Theorem 2.54, Corollary 2.42, and the equations (j′_{0}(A ⊗ B)) ∘ [Id_{V} ⊗ l_{U}] = [A ⊗ (B ∘ l_{U})] = [(A ∘ l_{V}) ⊗ B],

(T r_{V;U,W ⊗W′}(j′_{0}(A ⊗ B))) ∘ l_{U} = T r_{V;𝕂⊗U,W ⊗W′}([A ⊗ (B ∘ l_{U})])
= T r_{V;𝕂⊗U,W ⊗W′}([(A ∘ l_{V}) ⊗ B])
= j_{4}((T r_{V;𝕂,W} (A ∘ l_{V})) ⊗ B)
⟹ T r_{V;U,W ⊗W′}(j′_{0}(A ⊗ B)) = (j_{4}((T r_{V;𝕂,W} (A ∘ l_{V})) ⊗ B)) ∘ l_{U}^{−1} :
u ↦ ((T r_{V;𝕂,W} (A ∘ l_{V}))(1)) ⊗ (B(u))
= (T r_{V;W} (A)) ⊗ (B(u))
= (n′′((T r_{V;W} (A)) ⊗ B))(u).

Corollary 2.63. For finite-dimensional V_{1}, V_{2}, maps A : V_{1} → V_{1} ⊗ W_{1}, B : V_{2} → V_{2} ⊗ W_{2}, the switching map s_{1}, and a canonical j map, as in Theorem 2.40: j′_{3} : Hom(V_{1}, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2}, V_{2} ⊗ W_{2}) → Hom(V_{1} ⊗ V_{2}, V_{1} ⊗ V_{2} ⊗ W_{1} ⊗ W_{2}), the following identity holds:

T r_{V_{1}⊗V_{2};W_{1}⊗W_{2}}(s_{1} ∘ (j′_{3}(A ⊗ B))) = (T r_{V_{1};W_{1}}(A)) ⊗ (T r_{V_{2};W_{2}}(B)) ∈ W_{1} ⊗ W_{2}.

Proof. In the following diagram,

\xymatrix{M_{11} \ar[r]^{a_{1}} \ar[d]^{j′_{3}} & M_{12} \ar[r]^{j_{3}} & M_{13} \ar[d]^{a_{2}} \\ M_{21} \ar[r]^{a_{3}} & M_{22} \ar[r]^{a_{4}} & M_{23}}

the objects are

M_{11} = Hom(V_{1}, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2}, V_{2} ⊗ W_{2})
M_{21} = Hom(V_{1} ⊗ V_{2}, V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2})
M_{12} = Hom(V_{1} ⊗ 𝕂, V_{1} ⊗ W_{1}) ⊗ Hom(V_{2} ⊗ 𝕂, V_{2} ⊗ W_{2})
M_{22} = Hom(V_{1} ⊗ V_{2} ⊗ 𝕂, V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2})
M_{13} = Hom(V_{1} ⊗ 𝕂 ⊗ V_{2} ⊗ 𝕂, V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2})
M_{23} = Hom(V_{1} ⊗ V_{2} ⊗ 𝕂 ⊗ 𝕂, V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2}),

and the arrows are

a_{1} = [Hom(l_{V_{1}}, Id_{V_{1}⊗W_{1}}) ⊗ Hom(l_{V_{2}}, Id_{V_{1}⊗W_{2}})]
a_{2} = Hom(s_{2}, Id_{V_{1}⊗W_{1}⊗V_{2}⊗W_{2}})
a_{3} = Hom(l_{V_{1}⊗V_{2}}, Id_{V_{1}⊗W_{1}⊗V_{2}⊗W_{2}})
a_{4} = Hom([Id_{V_{1}⊗V_{2}} ⊗ l_{𝕂}], Id_{V_{1}⊗W_{1}⊗V_{2}⊗W_{2}}).

The diagram is commutative, by Lemma 1.37 and a scalar multiplication identity. By Theorem 2.54, Theorem 2.30, Theorem 2.40, and the diagram,

LHS = (T r_{V_{1}⊗V_{2};𝕂,W_{1}⊗W_{2}}(s_{1} ∘ (j′_{3}(A ⊗ B)) ∘ l_{V_{1}⊗V_{2}}))(1)
= (T r_{V_{1}⊗V_{2};𝕂,W_{1}⊗W_{2}}(s_{1}∘(j_{3}((A ∘ l_{V_{1}})⊗(B ∘ l_{V_{2}})))∘s_{2} ∘ [Id_{V_{1}⊗V_{2}} ⊗ l_{𝕂}^{−1}]))(1)
= ((T r_{V_{1}⊗V_{2};𝕂⊗𝕂,W_{1}⊗W_{2}}(s_{1} ∘ (j_{3}((A ∘ l_{V_{1}}) ⊗ (B ∘ l_{V_{2}}))) ∘ s_{2})) ∘ l_{𝕂}^{−1})(1)
= (j_{4}((T r_{V_{1};𝕂,W_{1}}(A ∘ l_{V_{1}})) ⊗ (T r_{V_{2};𝕂,W_{2}}(B ∘ l_{V_{2}}))))(1 ⊗ 1)
= (T r_{V_{1};W_{1}}(A)) ⊗ (T r_{V_{2};W_{2}}(B)) = RHS.

Example 2.64. If L is finite-dimensional and Ev_{L} : L^{∗} ⊗ L → 𝕂 is invertible, as in Lemma 2.21, then T r_{L;W} is invertible, because it is a composite of invertible maps by Definition 2.50:

T r_{L;W} = l_{W} ∘ [T r_{L} ⊗ Id_{W}] ∘ n^{−1} : Hom(L, L ⊗ W) → W.

Also by Lemma 2.21 and Example 2.51, for any w ∈ W,

T r_{L;W} (n(Id_{L} ⊗ w)) = 1 · w = w.

Exercise 2.65. For a direct sum V = V_{1} ⊕ V_{2} as in Definition 1.73, and maps A : V_{1} → V_{1} ⊗ W, B : V_{2} → V_{2} ⊗ W, define A ⊕ B : V → V ⊗ W, using the inclusions from Example 1.77:

A ⊕ B = [Q_{1} ⊗ Id_{W}] ∘ A ∘ P_{1} + [Q_{2} ⊗ Id_{W}] ∘ B ∘ P_{2}.

If V is finite-dimensional, then

T r_{V;W} (A ⊕ B) = T r_{V_{1};W} (A) + T r_{V_{2};W} (B).

Hint. The proof proceeds exactly as in Proposition 2.12, using Corollary 2.57.

Exercise 2.66. For V = V_{1} ⊕ V_{2}, and K : V → V ⊗ W,

T r_{V;W} (K) = T r_{V_{1};W} ([P_{1} ⊗ Id_{W}] ∘ K ∘ Q_{1}) + T r_{V_{2};W} ([P_{2} ⊗ Id_{W}] ∘ K ∘ Q_{2}).

Hint. Using Corollary 2.57 and Lemma 1.36,

T r_{V_{i};W} ([P_{i} ⊗ Id_{W}] ∘ K ∘ Q_{i}) = T r_{V;W} ([(Q_{i} ∘ P_{i}) ⊗ Id_{W}] ∘ K).

The proof proceeds exactly as in Proposition 2.13.

2.4. Equivalence of alternative definitions

In [JSV] §3, the canonical trace of a map F : V ⊗ U → V ⊗ W is defined in terms of category theory. In the context and notation of these notes, the definition of [JSV] can be interpreted as saying that T r_{V;U,W} (F) is the following composite map from U to W:

(2.8) \xymatrix{U \ar[r] & V^{∗} ⊗ V ⊗ U \ar[r]^{[Id_{V ^{∗}} ⊗F]} & V^{∗} ⊗ V ⊗ W \ar[r]^{l_{W} ∘[Ev_{V} ⊗Id_{W}]} & W,}

where the first arrow is defined for u ∈ U by:

u ↦ (k^{−1}(Id_{V})) ⊗ u.

As mentioned after Theorem 2.10, this map could be expressed in terms of maps l_{U} : 𝕂 ⊗ U → U and an inclusion Q_{1}^{1} : 𝕂 → End(V) : 1 ↦ Id_{V} as in Example 2.9 and Equation (2.4).

Notation 2.67. For finite-dimensional V, k : V^{∗} ⊗ V → End(V), a switching map s : V^{∗} ⊗ V → V ⊗ V^{∗}, and the map Q_{1}^{1} : 𝕂 → End(V) : 1 ↦ Id_{V}, define η_{V} : 𝕂 → V ⊗ V^{∗} by:

η_{V} = s ∘ k^{−1} ∘ Q_{1}^{1}.

The switching map is included for later convenience in Theorem 2.95. The arrow in (2.8) can then be described as follows:

[s^{−1} ⊗ Id_{U}] ∘ [η_{V} ⊗ Id_{U}] ∘ l_{U}^{−1}
(2.9)  = [k^{−1} ⊗ Id_{U}] ∘ [Q_{1}^{1} ⊗ Id_{U}] ∘ l_{U}^{−1} : U → V^{∗} ⊗ V ⊗ U
: u ↦ (k^{−1}(Id_{V})) ⊗ u.

The following Theorem shows that the formula (2.8) for T r_{V;U,W} (F) coincides with Definition 2.24. V must be finite-dimensional, but U and W may be arbitrary.

Theorem 2.68. For finite-dimensional V, F : V ⊗ U → V ⊗ W, and u ∈ U,

(T r_{V;U,W} (F))(u)=(l_{W} ∘ [Ev_{V} ⊗ Id_{W}] ∘ [Id_{V^{∗}} ⊗ F] ∘ [k^{−1}⊗ Id_{U}])(Id_{V} ⊗ u).

Proof. The following diagram is commutative, where the top arrow is a_{1} = [Id_{End(V^{∗})} ⊗ j_{2}].

[figure: End(V^{∗}) ⊗ Hom(V ⊗ U, V ⊗ W); j; End(V^{∗}) ⊗ End(V) ⊗ Hom(U, W); a_{1}; [j⊗Id_{Hom(U,W)}]; Hom(V^{∗} ⊗ V ⊗ U, V^{∗} ⊗ V ⊗ W); Hom([k^{−1}⊗Id_{U}],[Ev_{V} ⊗Id_{W}]); End(V^{∗} ⊗ V) ⊗ Hom(U, W); [Hom(k^{−1},Ev_{V} )⊗Id_{Hom(U,W)}]; j; Hom(End(V) ⊗ U, 𝕂 ⊗ W); Hom(Id_{End(V )⊗U} ,l_{W}); End(V)^{∗} ⊗ Hom(U, W); j; [(d_{End(V )}(Id_{V} ))⊗Id_{Hom(U,W)}]; Hom(End(V) ⊗ U, W); d_{End(V )⊗U,W} (Id_{V} ⊗u); 𝕂 ⊗ Hom(U, W); l_{1}; W; Hom(U, W); d_{UW} (u)]

The upper and lower squares are easy to check (the maps d_{UW}, d_{End(V)⊗U,W} are as in Definition 1.13), and the middle square is commutative by Lemma 1.37. Starting with Id_{V^{∗}} ⊗ F in the upper left corner, the RHS of the Theorem is the output of the composition in the left column. The top three arrows in the right column come from the construction in Theorem 2.10,

T r_{V} (A) = ((d_{End(V)}(Id_{V})) ∘ Hom(k^{−1}, Ev_{V}) ∘ j)(Id_{V^{∗}} ⊗ A),

so that the composite of a_{1}^{−1} = [Id_{End(V^{∗})} ⊗ j_{2}^{−1}] with the right column of maps takes Id_{V^{∗}} ⊗ F to T r_{V;U,W} (F) = l_{1}((j_{1}(T r_{V} ⊗ Id_{Hom(U,W)}))(j_{2}^{−1}(F))). The lowest arrow plugs u into T r_{V;U,W} (F), giving the LHS of the Theorem, so the equality follows directly from the commutativity of the diagram.

Corollary 2.69. For finite-dimensional V and A : V → V ⊗ W,

T r_{V;W} (A) = (l_{W} ∘ [Ev_{V} ⊗ Id_{W}] ∘ [Id_{V^{∗}} ⊗ A] ∘ k^{−1})(Id_{V}).

Proof. By Theorem 2.68 and Theorem 2.54,

LHS = (T r_{V;𝕂,W} (A ∘ l_{V}))(1)
= (l_{W} ∘ [Ev_{V} ⊗ Id_{W}] ∘ [Id_{V^{∗}} ⊗ (A ∘ l_{V})] ∘ [k^{−1} ⊗ Id_{𝕂}])(Id_{V} ⊗ 1)
= RHS.

This shows, in analogy with Theorem 2.10 and (2.8) from Theorem 2.68, that the W -valued trace of A is the output of the distinguished element k^{−1}(Id_{V}) under the composite map

(2.10)  \xymatrix{V^{∗} ⊗ V \ar[r]^{[Id_{V ^{∗}} ⊗A]} & V^{∗} ⊗ V ⊗ W \ar[r]^{l_{W} ∘[Ev_{V} ⊗Id_{W}]} & W.}

So, Corollary 2.69 could be used as an alternative, but equivalent, definition of vector valued trace. This Section continues with some identities for the vector valued trace, some of which (Theorem 2.73, Corollary 2.87, Corollary 2.107) could also be used in alternative approaches to the definition of T r_{V;W}.

2.4.1. A vector valued canonical evaluation.

Definition 2.70. The distinguished element

Ev_{V W} ∈ Hom(Hom(V, W) ⊗ V, W)

is the \underline{canonical evaluation} map, defined by

Ev_{V W} (A ⊗ v) = A(v).

In the W = 𝕂 case, Ev_{V 𝕂} is the distinguished element Ev_{V} ∈ (V^{∗} ⊗ V)^{∗} from Definition 2.2. The scalar evaluation Ev_{V} and vector valued evaluation Ev_{V W} are related as follows.

Lemma 2.71. For any V and W, let s′ : V ⊗ W → W ⊗ V be a switching map. The following diagram is commutative.

\xymatrix{V^{∗} ⊗ V ⊗ W \ar[r]_{[Ev_{V} ⊗Id_{W}]} \ar[d]^{[Id_{V ^{∗}} ⊗s′]} & 𝕂 ⊗ W \ar[dd]^{l} \\ V^{∗} ⊗ W ⊗ V \ar[d]^{[k_{V W} ⊗Id_{V} ]} \\ Hom(V, W) ⊗ V \ar[r]_{Ev_{V W}} & W}

Proof.

ϕ ⊗ v ⊗ w ↦ (l ∘ [Ev_{V} ⊗ Id_{W}])(ϕ ⊗ v ⊗ w)
= ϕ(v) · w,
ϕ ⊗ v ⊗ w ↦ (Ev_{V W} ∘ [k_{V W} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s′])(ϕ ⊗ v ⊗ w)
= Ev_{V W} ((k_{V W} (ϕ ⊗ w)) ⊗ v)
= ϕ(v) · w.

The canonical evaluation maps have the following naturality property.

Lemma 2.72. For any vector spaces U, V, V′, W, and any maps G : V′ → V, B : U → W, the following diagram is commutative.

\xymatrix{Hom(V, U) ⊗ V \ar[r]^{Ev_{V U}} & U \ar[dd]^{B} \\ Hom(V, U) ⊗ V′ \ar[u]^{[Id_{Hom(U,V )}⊗G]} \ar[d]_{[Hom(G,B)⊗Id_{V ′}]} \\ Hom(V′, W) ⊗ V′ \ar[r]^{Ev_{V ′W}} & W}

Proof. For A ∈ Hom(V, U), v ∈ V′,

B ∘ Ev_{V U} ∘ [Id_{Hom(V,U)} ⊗ G] :
A ⊗ v ↦ B(Ev_{V U}(A ⊗ (G(v)))) = B(A(G(v))),
Ev_{V′W} ∘ [Hom(G, B) ⊗ Id_{V′}] :
A ⊗ v ↦ Ev_{V′W} ((B ∘ A ∘ G) ⊗ v) = (B ∘ A ∘ G)(v).

Theorem 2.73. For a map

n′ : Hom(V, W) ⊗ V → Hom(V, V ⊗ W),

if V is finite-dimensional, then

T r_{V;W} ∘ n′ = Ev_{V W} .

Proof. The map n′ is as in Notation 1.40. The conclusion is equivalent to the formula T r_{V;W} (n′(A ⊗ v)) = A(v), for A ∈ Hom(V, W) and v ∈ V.

The claimed equality T r_{V;W} ∘ n′ = Ev_{V W} appears in the lower triangle in the following diagram, so it follows from the commutativity of the rest of the diagram.

[figure: End(V) ⊗ W; [T r_{V} ⊗Id_{W}]; n; V^{∗} ⊗ V ⊗ W; [k_{V V} ⊗Id_{W}]; [Ev_{V} ⊗Id_{W}]; [Id_{V ^{∗}} ⊗s′]; k_{V,V ⊗W}; 𝕂 ⊗ W; l; V^{∗} ⊗ W ⊗ V; [k_{V W} ⊗Id_{V} ]; Hom(V, V ⊗ W); T r_{V ;W}; Hom(V, W) ⊗ V; n′; Ev_{V W}; W]

The front left square is Definition 2.50, and the upper triangle is Definition 2.3, together with Lemma 1.36. The front right square is exactly Lemma 2.71. The back square is commutative, where the back left triangle is exactly Lemma 1.61, and the back right triangle is a variation on Lemma 1.61, with an extra switching s′ and the differently ordered n′, checked by the following calculation.

(2.11)  ϕ ⊗ v ⊗ w ↦ (n′ ∘ [k_{V W} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s′])(ϕ ⊗ v ⊗ w)
= n′((k_{V W} (ϕ ⊗ w)) ⊗ v) :
u ↦ v ⊗ (ϕ(u) · w) = (k_{V,V ⊗W} (ϕ ⊗ v ⊗ w))(u).

Recall the canonical map e_{V V}^{W} : End(V) → Hom(Hom(V, W)⊗V, W) from Definition 1.52. In the following diagram, the left triangle is commutative by Lemma 1.54.

\xymatrix{ & Hom(Hom(V, W) ⊗ V, W) \\ End(V) \ar[ur]^{e_{V V}^{W}} \ar[d]_{t_{V V}^{W}} &  & Hom(Hom(V, V ⊗ W), W) \ar[ul]_{Hom(n′,Id_{W})} \\ End(Hom(V, W)) \ar[uur]_{q}}

Every space in the diagram contains a distinguished element, giving an analogue of Equation (2.1):

(2.12)  Ev_{V W} = q(Id_{Hom(V,W)}) = q(t_{V V}^{W} (Id_{V})) = e_{V V}^{W} (Id_{V}).

Theorem 2.73 gives this analogue of Equation (2.2) from Lemma 2.5:

T r_{V;W} = ((Hom(n′, Id_{W}))^{−1} ∘ q)(Id_{Hom(V,W)}) = ((Hom(n′, Id_{W}))^{−1} ∘ e_{V V}^{W})(Id_{V}).

Using Theorem 2.73 as a definition for the vector valued trace allows some proofs to be simplified and avoids scalar multiplication. For example, the following result re-states Corollary 2.58 but gives a simpler proof.

Corollary 2.74. For any V, W, W′, if V is finite-dimensional and B ∈ Hom(W, W′) then

T r_{V;W′} ∘ Hom(Id_{V} , [Id_{V} ⊗ B]) = B ∘ T r_{V;W} : Hom(V, V ⊗ W) → W′.

Proof. For n′ as in Theorem 2.73 and an analogous map n′′, the downward composite in the left column of the following diagram is T r_{V;W} = Ev_{V W} ∘ (n′)^{−1}, and in the right column is T r_{V;W′}.

\xymatrix{Hom(V, V ⊗ W) \ar[r]^{Hom(Id_{V} ,[Id_{V} ⊗B])} & Hom(V, V ⊗ W′) \\ Hom(V, W) ⊗ V \ar[u]^{n′} \ar[r]^{[Hom(Id_{V} ,B)⊗Id_{V} ]} \ar[d]_{Ev_{V W}} & Hom(V, W′) ⊗ V \ar[u]_{n′′} \ar[d]^{Ev_{V W′}} \\ W \ar[r]^{B} & W′}

The blocks are commutative; the upper by Lemma 1.41 and the lower by Lemma 2.72 (in the case G = Id_{V}).

Theorem 2.75. For finite-dimensional V, let

n_{1} : End(V) ⊗ U → Hom(V, V ⊗ U).

Then, for any F : V ⊗ U → V ⊗ W and u ∈ U,

(T r_{V;U,W} (F))(u) = T r_{V;W} (F ∘ (n_{1}(Id_{V} ⊗ u))).

Proof. Consider the following diagram.

[figure: U; V^{∗}⊗V ⊗U; [k⊗Id_{U}]; [Id_{V ^{∗}} ⊗F]; k_{V,V ⊗U}; V^{∗}⊗V ⊗W; k_{V,V ⊗W}; [Id_{V ^{∗}} ⊗s′]; [Ev_{V} ⊗Id_{W}]; 𝕂⊗W; l_{W}; 𝕂 ⊗ U; l_{U}; [Q_{1}^{1}⊗Id_{U}]; End(V) ⊗ U; n_{1}; V^{∗}⊗W ⊗V; [k_{V W} ⊗Id_{V} ]; Hom(V, V⊗U); Hom(Id_{V},F); Hom(V,V⊗W); Hom(V,W)⊗V; n′; Ev_{V W}; W]

The composition from U to W clockwise along the upper row gives T r_{V;U,W} (F) by Theorem 2.68. The left square is from (2.9), and the right block is Lemma 2.71. The center left triangle with the n_{1} map is exactly Lemma 1.61, and the center right triangle with the n′ map is the variation on Lemma 1.61 copied from (2.11) in the Proof of Theorem 2.73. The center block is commutative by Lemma 1.58. The claim follows from Theorem 2.73:

LHS = Ev_{V W} ((n′)^{−1}(F ∘ (n_{1}(Id_{V} ⊗ u)))) = RHS.

Corollary 2.69 and Theorem 2.73 are related by the following commutative diagram.

\xymatrix{V^{∗} ⊗ V \ar[r]^{[Id_{V ^{∗}} ⊗A]} \ar[d]^{k} & V^{∗} ⊗ V ⊗ W \ar[d]^{k_{V,V ⊗W}} \ar[r]^{[Ev_{V} ⊗Id_{W}]} & 𝕂 ⊗ W \ar[r]^{l_{W}} & W \\ End(V) \ar[r]^{Hom(Id_{V} ,A)} & Hom(V, V ⊗ W) & Hom(V, W) ⊗ V \ar[l]^{n′} \ar[ur]^{Ev_{V W}}}

The left square is commutative by Lemma 1.58, and the right block is from the Proof of Theorem 2.75. Starting with Id_{V} ∈ End(V), the composition clockwise along the upper row gives T r_{V;W} (A) ∈ W as in (2.10) from Corollary 2.69, and counterclockwise along the lower row gives Ev_{V W} ((n′)^{−1}(A)), which also equals T r_{V;W} (A) by Theorem 2.73.

Lemma 2.76. For any U, V, W, the following diagram is commutative.

\xymatrix{U ⊗ Hom(V, W) ⊗ V \ar[d]_{[n⊗Id_{V} ]} \ar[r]^{[Id_{U} ⊗Ev_{V W}]} & U ⊗ W \\ Hom(V, U ⊗ W) ⊗ V \ar[ur]_{Ev_{V,U⊗W}}}

Proof. Both paths take u ⊗ A ⊗ v ∈ U ⊗ Hom(V, W) ⊗ V to u ⊗ (A(v)) ∈ U ⊗ W.

Theorem 2.77. For any V, U, W, and F : V ⊗ U → V ⊗ W, if V is finitedimensional then the n maps in the following diagram are invertible:

\xymatrix{V ⊗ U \ar[r]^{F} & V ⊗ W \\ V ⊗ Hom(V, U) ⊗ V \ar[d]_{[n_{2}⊗Id_{V} ]} \ar[u]^{[Id_{V} ⊗Ev_{V U}]} & V ⊗ Hom(V, W) ⊗ V \ar[d]^{[n′_{2}⊗Id_{V} ]} \ar[u]_{[Id_{V} ⊗Ev_{V W}]} \\ Hom(V, V ⊗ U) ⊗ V \ar[r]_{[Hom(Id_{V} ,F)⊗Id_{V} ]} & Hom(V, V ⊗ W) ⊗ V}

and the diagram is commutative, in the sense that

F ∘ [Id_{V} ⊗ Ev_{V U}] ∘ [n_{2} ⊗ Id_{V}]^{−1}
= [Id_{V} ⊗ Ev_{V W}] ∘ [n′_{2} ⊗ Id_{V}]^{−1} ∘ [Hom(Id_{V} , F) ⊗ Id_{V}].

Proof. The n_{2}, n′_{2} maps are defined as they appear in the diagram, with subscript notation used to avoid duplication with the labels appearing elsewhere in this Section; they are invertible by Lemma 1.43.

By Lemma 2.76, the upward composite on the left, [Id_{V} ⊗Ev_{V U}]∘[n_{2} ⊗Id_{V}]^{−1}, is equal to Ev_{V,V ⊗U}, and similarly the upward composite on the right is equal to Ev_{V,V ⊗W}. The claim then follows from Lemma 2.72.

Corollary 2.78. For any V, U, W, and F : V ⊗ U → V ⊗ W, if V is finite-dimensional then the n maps in the following diagram are invertible:

\xymatrix{V ⊗ U \ar[r]^{F} & V ⊗ W \\ V ⊗ Hom(V, U) ⊗ V \ar[d]_{[Id_{V} ⊗n_{3}]} \ar[u]^{[Id_{V} ⊗Ev_{V U}]} & V ⊗ Hom(V, W) ⊗ V \ar[d]^{[Id_{V} ⊗n′_{3}]} \ar[u]_{[Id_{V} ⊗Ev_{V W}]} \\ V ⊗ Hom(V, U ⊗ V) \ar[d]_{n_{4}} & V ⊗ Hom(V, W ⊗ V) \ar[d]^{n′_{4}} \\ Hom(V, V ⊗ U ⊗ V) \ar[r]_{Hom(Id_{V} ,[F ⊗Id_{V} ])} & Hom(V, V ⊗ W ⊗ V)}

and the diagram is commutative, in the sense that

F ∘ [Id_{V} ⊗ Ev_{V U}] ∘ [Id_{V} ⊗ n_{3}]^{−1} ∘ n_{4}^{−1}
= [Id_{V} ⊗ Ev_{V W}] ∘ [Id_{V} ⊗ n′_{3}]^{−1} ∘ (n′_{4})^{−1} ∘ Hom(Id_{V} , [F ⊗ Id_{V}]).

Proof. The four n maps are defined as they appear in the diagram, with subscript notation used to avoid duplication with the labels appearing elsewhere in this Section; they are invertible by Lemma 1.43. The left column of the following diagram matches the left column from the diagram in the Theorem; the notation V = V_{1} = V_{2} = V_{3} is introduced to track the action of the switching maps.

\xymatrix{V_{1} ⊗ U \\  & V_{1} ⊗ 𝕂 ⊗ U \ar[ul]^{[Id_{V} ⊗l_{U}]} \ar[r]_{[s_{1}⊗Id_{U}]} & 𝕂 ⊗ V_{1} ⊗ U \ar@{-->}[ull]_{l_{V ⊗U}} \\ V_{1} ⊗ Hom(V_{3}, U) ⊗ V_{2} \ar[dd]_{[Id_{V} ⊗n_{3}]} \ar[uu]^{[Id_{V} ⊗Ev_{V U}]} & V_{1} ⊗ V_{3}^{∗} ⊗ V_{2} ⊗ U \ar[d]^{[Id_{V} ⊗[Id_{V ^{∗}} ⊗s′]]} \ar[r]_{[s_{2}⊗Id_{U}]} \ar[u]^{[Id_{V} ⊗[Ev_{V} ⊗Id_{U}]]} & V_{3}^{∗} ⊗ V_{2} ⊗ V_{1} ⊗ U \ar[dddl]^{[Id_{V ^{∗}} ⊗s_{3}]} \ar[u]_{[Ev_{V} ⊗Id_{V ⊗U}]} \\  & V_{1} ⊗ V_{3}^{∗} ⊗ U ⊗ V_{2} \ar[dd]_{[s_{4}⊗Id_{U⊗V} ]} \ar[ul]_{[Id_{V} ⊗[k_{V U} ⊗Id_{V} ]]} \ar[dl]_{[Id_{V} ⊗k_{V,U⊗V} ]} \\ V_{1} ⊗ Hom(V_{3}, U ⊗ V_{2}) \ar[d]_{n_{4}} \\ Hom(V_{3}, V_{1} ⊗ U ⊗ V_{2}) & V_{3}^{∗} ⊗ V_{1} ⊗ U ⊗ V_{2} \ar[l]_{k_{V,V ⊗U⊗V}}}

The commutativity of the blocks on the right is easily checked, and the upper left block is commutative by Lemma 1.36 and Lemma 2.71. The center left and lower left blocks are variations on Lemma 1.61; to check the lower left, with v, w, x ∈ V, ϕ ∈ V^{∗}, u ∈ U:

v ⊗ ϕ ⊗ u ⊗ w ↦ k_{V,V ⊗U⊗V} (ϕ ⊗ v ⊗ u ⊗ w) :
x ↦ ϕ(x) · v ⊗ u ⊗ w,
v ⊗ ϕ ⊗ u ⊗ w ↦ n_{4}(v ⊗ (k_{V,V ⊗U}(ϕ ⊗ u ⊗ w))) :
x ↦ v ⊗ (ϕ(x) · u ⊗ w).

So, the diagram is commutative, and a similar result holds when U is replaced by W, corresponding to the right column of the diagram of the Theorem. The claim of the Theorem is reduced to the commutativity of the following diagram.

\xymatrix{V_{1} ⊗ U \ar[r]^{F} & V_{1} ⊗ W \\ 𝕂 ⊗ V_{1} ⊗ U \ar[u]^{l_{V ⊗U}} \ar[r]^{[Id_{𝕂}⊗F]} & 𝕂 ⊗ V_{1} ⊗ W \ar[u]^{l_{V ⊗W}} \\ V_{3}^{∗} ⊗ V_{2} ⊗ V_{1} ⊗ U \ar[d]_{[Id_{V ^{∗}} ⊗s_{3}]} \ar[u]^{[Ev_{V} ⊗Id_{V ⊗U}]} \ar[r]^{[Id_{V ^{∗}⊗V} ⊗F]} & V_{3}^{∗} ⊗ V_{2} ⊗ V_{1} ⊗ W \ar[d]^{[Id_{V ^{∗}} ⊗s′_{3}]} \ar[u]_{[Ev_{V} ⊗Id_{V ⊗W}]} \\ V_{3}^{∗} ⊗ V_{1} ⊗ U ⊗ V_{2} \ar[d]_{k_{V,V ⊗U⊗V}} \ar[r]^{[Id_{V}^{∗} ⊗[F ⊗Id_{V} ]]} & V_{3}^{∗} ⊗ V_{1} ⊗ W ⊗ V_{2} \ar[d]^{k_{V,V ⊗W ⊗V}} \\ Hom(V_{3}, V_{1} ⊗ U ⊗ V_{2}) \ar[r]^{Hom(Id_{V} ,[F ⊗Id_{V} ])} & Hom(V_{3}, V_{1} ⊗ W ⊗ V_{2})}

The upper block is commutative by Lemma 1.38, the next lower block is commutative by Lemma 1.36, and the lowest block is commutative by Lemma 1.58. The remaining block is easily checked, the k maps are invertible by Lemma 1.60, and the claim follows.

Lemma 2.79. For a switching map s : V ⊗V → V ⊗V, if k : V^{∗} ⊗V → End(V) is invertible, then the following map:

v ↦ (l_{V} ∘ [Ev_{V} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s] ∘ [k^{−1} ⊗ Id_{V}])(Id_{V} ⊗ v)

is equal to the identity map Id_{V}.

Proof. For the special case of Lemma 2.71 with W = V, s′ = s, the above composite map is Ev_{V V}, so the given expression is

v ↦ Ev_{V V} (Id_{V} ⊗ v) = Id_{V} (v) = v.

Example 2.80. If V is finite-dimensional, then the generalized trace of the switching map s : V ⊗ V → V ⊗ V is:

(2.13)  T r_{V;V,V} (s) = Id_{V} ,

by the formula from Theorem 2.68 and Lemma 2.79:

T r_{V;V,V} (s) : v ↦ (l_{V} ∘ [Ev_{V} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s] ∘ [k^{−1}⊗ Id_{V}])(Id_{V} ⊗ v) = v.

Remark 2.81. Equation (2.13) is related to the “yanking” property of [JSV].

The following Lemma is analogous to Lemma 2.79.

Lemma 2.82. For a switching involution

s′′ : V^{∗} ⊗ V ⊗ V^{∗} → V^{∗} ⊗ V ⊗ V^{∗} : ϕ ⊗ v ⊗ ψ ↦ ψ ⊗ v ⊗ ϕ,

if k : V^{∗} ⊗ V → End(V) is invertible, then the following map:

ϕ ↦ (l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V}] ∘ s′′ ∘ [k^{−1} ⊗ Id_{V^{∗}}])(Id_{V} ⊗ ϕ)

is equal to the identity map Id_{V^{∗}}.

Proof. The following diagram is commutative; the calculation is similar that in the Proof of Lemma 2.71. Abbreviate t = t_{V V} as in Lemma 2.5.

\xymatrix{V^{∗} ⊗ V ⊗ V^{∗} \ar[r]_{[Ev_{V} ⊗Id_{V ^{∗}}]} \ar[d]^{s′′} & 𝕂 ⊗ V^{∗} \ar[ddd]^{l_{V ^{∗}}} \\ V^{∗} ⊗ V ⊗ V^{∗} \ar[d]^{[k⊗Id_{V ^{∗}}]} \\ End(V) ⊗ V^{∗} \ar[d]^{[t⊗Id_{V ^{∗}}]} \\ End(V^{∗}) ⊗ V^{∗} \ar[r]_{Ev_{V ^{∗}V ^{∗}}} & V^{∗}}

Starting with Id_{V} ⊗ ϕ, the commutativity of the diagram and the existence of k^{−1} give:

(l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V}] ∘ s′′ ∘ [k^{−1} ⊗ Id_{V^{∗}}])(Id_{V} ⊗ ϕ) (2.14)

= Ev_{V^{∗}V^{∗}}((t(Id_{V})) ⊗ ϕ) = Id_{V}^{∗} (ϕ) = ϕ.

Theorem 2.83. If k : V^{∗} ⊗ V → End(V) is invertible, then d_{V} : V → V^{∗∗} is invertible.

Proof. The following map, temporarily denoted B : V^{∗∗} → V, is an inverse:

B : Φ ↦ l_{V} ([Φ ⊗ Id_{V}](k^{−1}(Id_{V}))).

For any V, W, and v ∈ V, the following diagram is commutative (the two paths V^{∗} ⊗ W → W are equal compositions).

\xymatrix{V^{∗} ⊗ W \ar[r]^{k_{V W}} \ar[d]_{[(d_{V} (v))⊗Id_{W}]} & Hom(V, W) & Hom(V, W) ⊗ 𝕂 \ar[l]_{l} \ar[d]^{[Id_{Hom(V,W)}⊗(m(v))]} \\ 𝕂 ⊗ W \ar[r]^{l_{W}} & W & Hom(V, W) ⊗ V \ar[l]_{Ev_{V W}}}

In the case W = V, starting with Id_{V} in the top middle gives the following equality:

(B ∘ d_{V})(v) = l_{V} ([(d_{V} (v)) ⊗ Id_{V}](k^{−1}(Id_{V})))
= Ev_{V V} (Id_{V} ⊗ v) = v.

To check the composite in the other order, in the second diagram, s′′′ is another switching map as indicated in the diagram, η_{V} and s are as in (2.9), the block is commutative, and the composition in the left column acts as the identity map, by (2.14) from Lemma 2.82.

\xymatrix{V^{∗} \ar[d]^{l_{V ^{∗}}^{−1}} \\ 𝕂 ⊗ V^{∗} \ar[d]^{[η_{V} ⊗Id_{V ^{∗}}]} \\ V ⊗ V^{∗} ⊗ V^{∗} \ar[d]^{[s⊗Id_{V ^{∗}}]} \\ V^{∗} ⊗ V ⊗ V^{∗} \ar[d]^{s′′} \ar[r]^{[[Φ⊗Id_{V} ]⊗Id_{V ^{∗}}]} & 𝕂 ⊗ V ⊗ V^{∗} \ar[d]^{s′′′} \\ V^{∗} ⊗ V ⊗ V^{∗} \ar[d]^{[Ev_{V} ⊗Id_{V ^{∗}}]} & V^{∗} ⊗ 𝕂 ⊗ V \ar[d]^{[Id_{V ^{∗}} ⊗l_{V} ]} \\ 𝕂 ⊗ V^{∗} \ar[d]^{l_{V ^{∗}}} & V^{∗} ⊗ V \ar[d]^{Ev_{V}} \\ V^{∗} \ar[r]^{Φ} & 𝕂}

The conclusion is:

Φ(ϕ) = (Ev_{V} ∘ [Id_{V^{∗}} ⊗ l_{V}] ∘ s′′′ ∘ [[Φ ⊗ Id_{V}] ⊗ Id_{V^{∗}}])((k^{−1}(Id_{V})) ⊗ ϕ)
= Ev_{V} (ϕ ⊗ (l_{V} ([Φ ⊗ Id_{V}](k^{−1}(Id_{V})))))
= ϕ(l_{V} ([Φ ⊗ Id_{V}](k^{−1}(Id_{V}))))
= ϕ(B(Φ)) = ((d_{V} ∘ B)(Φ))(ϕ).

Proposition 2.84. Given U, V, W, a map G : U → Hom(V, W), and the canonical map

q : Hom(U, Hom(V, W)) → Hom(U ⊗ V, W),

if V is finite-dimensional then

(2.15)  q(G) = T r_{V;W} ∘ n′ ∘ [G ⊗ Id_{V}].

Proof. q is as in Definition 1.44, and n′ is as in Theorem 2.73, which gives, for u ⊗ v ∈ U ⊗ V,

(T r_{V;W} ∘ n′ ∘ [G ⊗ Id_{V}])(u ⊗ v) = (Ev_{V W} ∘ [G ⊗ Id_{V}])(u ⊗ v)
= (G(u))(v)
= (q(G))(u ⊗ v).

Equation (2.15) from Proposition 2.84 can also be re-written, for F = q(G) ∈ Hom(U ⊗ V, W),

F = Ev_{V W} ∘ [(q^{−1}(F)) ⊗ Id_{V}] = T r_{V;W} ∘ n′ ∘ [(q^{−1}(F)) ⊗ Id_{V}].

Theorem 2.85. For finite-dimensional V, and maps

n_{2} : Hom(U, Hom(V, W)) ⊗ V → Hom(U, Hom(V, W) ⊗ V)
n_{3} : Hom(V ⊗ U, W) ⊗ V → Hom(V ⊗ U, V ⊗ W)
q : Hom(U, Hom(V, W)) → Hom(V ⊗ U, W)
F : V ⊗ U → V ⊗ W,

the following diagram is commutative.

\xymatrix{U \ar[r]^{T r_{V ;U,W} (F)} \ar[dr]_{(n_{2}∘[q^{−1}⊗Id_{V} ]∘n_{3}^{−1})(F)} & W \\  & Hom(V, W) ⊗ V \ar[u]_{Ev_{V W}}}

Proof. n_{3} is invertible by Lemma 1.43. The claim follows from the commutativity of this diagram.

\xymatrix{Hom(V ⊗ U, V ⊗ W) \ar[dd]_{T r_{V ;U,W}} & Hom(V ⊗ U, W) ⊗ V \ar[l]^{n_{3}} \\  & Hom(U, Hom(V, W)) ⊗ V \ar[d]^{n_{2}} \ar[u]_{[q⊗Id_{V} ]} \\ Hom(U, W) & Hom(U, Hom(V, W) ⊗ V) \ar[l]^{Hom(Id_{U} ,Ev_{V W})}}

Starting with A ⊗ v_{1} ∈ Hom(U, Hom(V, W)) ⊗ V,

(Hom(Id_{V} , Ev_{V W}) ∘ n_{2})(A ⊗ v_{1}) : u ↦ Ev_{V W} ((n_{2}(A ⊗ v_{1}))(u))
= Ev_{V W} ((A(u)) ⊗ v_{1}) = (A(u))(v_{1}).

Going the other way around the diagram,

T r_{V;U,W} (n_{3}((q(A)) ⊗ v_{1})) : u ↦ T r_{V;W} ((n_{3}((q(A)) ⊗ v_{1})) ∘ (n_{1}(Id_{V} ⊗ u)))
= T r_{V;W} (n′((A(u)) ⊗ v_{1}))
= (A(u))(v_{1}).

The first step uses Theorem 2.75 and its n_{1} map, and the last step uses Theorem 2.73 and its n′ map. The middle step uses the following calculation:

(n_{3}((q(A)) ⊗ v_{1})) ∘ (n_{1}(Id_{V} ⊗ u)) : v_{2} ↦ (n_{3}((q(A)) ⊗ v_{1}))(v_{2} ⊗ u)
= v_{1} ⊗ ((q(A))(v_{2} ⊗ u))
= v_{1} ⊗ ((A(u))(v_{2}))
= (n′((A(u)) ⊗ v_{1}))(v_{2}).

Theorem 2.86. For finite-dimensional V and any vector space Z, let s_{1} : V ⊗ V → V ⊗ V be the switching map and let

n : End(V) ⊗ V ⊗ Z → Hom(V, V ⊗ V ⊗ Z).

Then, for any B ∈ V ⊗ Z,

T r_{V;V ⊗Z}([s_{1} ⊗ Id_{Z}] ∘ (n(Id_{V} ⊗ B))) = B.

Proof. The following diagram is commutative, the lower triangle by Lemma 1.61 and the upper block in a variation on Lemma 1.61 that is straightforward to check.

\xymatrix{End(V) ⊗ V ⊗ Z \ar[d]_{n} & V^{∗} ⊗ V ⊗ V ⊗ Z \ar[l]_{[k_{V V} ⊗Id_{V ⊗Z}]} \ar[d]^{[[Id_{V ^{∗}} ⊗s_{1}]⊗Id_{Z}]} \\ Hom(V, V ⊗ V ⊗ Z) \ar[d]_{Hom(Id_{V} ,[s_{1}⊗Id_{Z}])} & V^{∗} ⊗ V ⊗ V ⊗ Z \ar[dl]^{k_{V,V ⊗V ⊗Z}} \ar[d]^{[k_{V V} ⊗Id_{V ⊗Z}]} \\ Hom(V, V ⊗ V ⊗ Z) & End(V) ⊗ V ⊗ Z \ar[l]^{n}}

So the LHS of the claim is:

T r_{V,V ⊗Z}([s_{1} ⊗ Id_{Z}] ∘ (n(Id_{V} ⊗ B)))
= (T r_{V;V ⊗Z} ∘ Hom(Id_{V} , [s_{1} ⊗ Id_{Z}]) ∘ n)(Id_{V} ⊗ B)
= (T r_{V;V ⊗Z} ∘ n ∘ [k_{V V} ⊗ Id_{V ⊗Z}]
∘[[Id_{V^{∗}} ⊗ s_{1}] ⊗ Id_{Z}] ∘ [k_{V V}^{−1} ⊗ Id_{V ⊗Z}])(Id_{V} ⊗ B).

Let s_{3} : V ⊗ Z → Z ⊗ V be another switching, and denote V_{1} = V_{2} = V, so that the labels explain the switching

s_{2} = [Id_{V^{∗}} ⊗ s_{4}] : V^{∗} ⊗ V_{1} ⊗ (V_{2} ⊗ Z) → V^{∗} ⊗ (V_{2} ⊗ Z) ⊗ V_{1}.

Using W = V ⊗ Z and this s_{4} in the role of s′ from the diagram from the Proof of Theorem 2.73, the commutativity of that diagram continues the chain of equalities:

= (Ev_{V,V ⊗Z} ∘ [k_{V,V ⊗Z} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s_{4}]
∘[[Id_{V^{∗}} ⊗ s_{1}] ⊗ Id_{Z}] ∘ [k_{V V}^{−1} ⊗ Id_{V ⊗Z}])(Id_{V} ⊗ B)
= (Ev_{V,V ⊗Z} ∘ [k_{V,V ⊗Z} ⊗ Id_{V}] ∘ [Id_{V^{∗}⊗V} ∘ s_{3}] ∘ [k_{V V}^{−1} ⊗ Id_{V ⊗Z}])(Id_{V} ⊗ B)
= (Ev_{V,V ⊗Z} ∘ [k_{V,V ⊗Z} ⊗ Id_{V}] ∘ [k_{V V}^{−1} ∘ s_{3}])(Id_{V} ⊗ B).

The commutativity of the following diagram:

\xymatrix{V^{∗} ⊗ V ⊗ Z ⊗ V \ar[d]_{[k_{V V} ⊗s_{3}^{−1}]} \ar[r]^{[k_{V,V ⊗Z}⊗Id_{V} ]} & Hom(V, V ⊗ Z) ⊗ V \ar[d]^{Ev_{V,V ⊗Z}} \\ End(V) ⊗ V ⊗ Z \ar[r]^{[Ev_{V,V} ⊗Id_{Z}]} & V ⊗ Z}

leads the equalities to the conclusion:

= [Ev_{V,V} ⊗ Id_{Z}](Id_{V} ⊗ B) = B.

The switching s_{3} : V ⊗ Z → Z ⊗ V from Theorem 2.86 appears in the following Corollaries, which are related to constructions in [Stolz-Teichner].

Corollary 2.87. Given V finite-dimensional and a map A : V → V ⊗ W, if there exist a vector space Z and a factorization of the form

A = [Id_{V} ⊗ B_{2}] ∘ [B_{1} ⊗ Id_{V}] ∘ l^{−1},

for l : 𝕂 ⊗ V → V, B_{1} : 𝕂 → V ⊗ Z, and B_{2} : Z ⊗ V → W, then

T r_{V;W} (A) = (B_{2} ∘ s_{3} ∘ B_{1})(1).

Proof. It is straightforward to check that the following diagram is commutative. The s_{1} is the switching from Theorem 2.86, and as in the above Proof, denote V_{1} = V_{2} = V, to keep track of switching, so that s_{4} : V_{1} ⊗(V_{2} ⊗Z) → (V_{2} ⊗Z)⊗V_{1}.

\xymatrix{V_{1} \ar[dd]_{n(Id_{V} ⊗(B_{1}(1)))} & 𝕂 ⊗ V_{1} \ar[d]^{[B_{1}⊗Id_{V} ]} \ar[l]^{l} \\  & V_{2} ⊗ Z ⊗ V_{1} \\ V_{1} ⊗ V_{2} ⊗ Z \ar[ur]^{s_{4}} \ar[dr]_{[s_{1}⊗Id_{Z}]} \\  & V_{2} ⊗ V_{1} ⊗ Z \ar[uu]_{[Id_{V} ⊗s_{3}]}}

The following equalities use the commutativity of the diagram, Corollary 2.58 (or Corollary 2.74) twice, and Theorem 2.86 applied to B = B_{1}(1) ∈ V ⊗ Z.

T r_{V;W} (A) = T r_{V;W} ([Id_{V} ⊗ B_{2}] ∘ [B_{1} ⊗ Id_{V}] ∘ l^{−1})
= B_{2}(T r_{V;Z⊗V} ([B_{1} ⊗ Id_{V}] ∘ l^{−1}))
= B_{2}(T r_{V;Z⊗V} ([Id_{V} ⊗ s_{3}] ∘ [s_{1} ⊗ Id_{Z}] ∘ (n(Id_{V} ⊗ (B_{1}(1))))))
= (B_{2} ∘ s_{3})(T r_{V;V ⊗Z}([s_{1} ⊗ Id_{Z}] ∘ (n(Id_{V} ⊗ (B_{1}(1))))))
= (B_{2} ∘ s_{3})(B_{1}(1)).

Corollary 2.88. Given V finite-dimensional and a map A : V → V, if there exist a vector space Z and a factorization of the form

A = l_{V} ∘ [Id_{V} ⊗ B_{2}] ∘ [B_{1} ⊗ Id_{V}] ∘ l^{−1},

for l_{V} : V ⊗ 𝕂 → V, l : 𝕂 ⊗ V → V, B_{1} : 𝕂 → V ⊗ Z, and B_{2} : Z ⊗ V → 𝕂, then

T r_{V} (A) = (B_{2} ∘ s_{3} ∘ B_{1})(1).

Proof. Theorem 2.53 and Corollary 2.87 apply.

Corollary 2.89. Given V finite-dimensional and a map A : V ⊗U → V ⊗W, if there exist a vector space Z and a factorization of the form

A = [Id_{V} ⊗ B_{2}] ∘ [B_{1} ⊗ Id_{V}] ∘ s_{5},

for s_{5} : V ⊗ U → U ⊗ V, B_{1} : U → V ⊗ Z, and B_{2} : Z ⊗ V → W, then

T r_{V;U,W} (A) = B_{2} ∘ s_{3} ∘ B_{1}.

Proof. Theorem 2.30 applies:

T r_{V;U,W} (A) = T r_{V;U,W} ([Id_{V} ⊗ B_{2}] ∘ [B_{1} ⊗ Id_{V}] ∘ s_{5})
= B_{2} ∘ (T r_{V;U,Z⊗V} ([B_{1} ⊗ Id_{V}] ∘ s_{5})),

so to prove the claim it is enough to check

(2.16)  T r_{V;U,Z⊗V} ([B_{1} ⊗ Id_{V}] ∘ s_{5}) = s_{3} ∘ B_{1}

for B_{1} ∈ Hom(U, V ⊗ Z). In the following diagram, temporarily denote by a_{1} the following map,

a_{1} = j ∘ [Id_{Hom(U,V ⊗Z)} ⊗ Q_{1}^{1}] ∘ l_{Hom(U,V ⊗Z)}^{−1} :
B_{1} ↦ [B_{1} ⊗ Id_{V}],

for Q_{1}^{1} : 𝕂 → End(V) as in Example 2.9.

\xymatrix{V ⊗ Hom(U, Z) \ar[r]^{s′_{3}} \ar[d]_{n′′} & Hom(U, Z) ⊗ V \ar[d]^{n_{1}} \\ Hom(U, V ⊗ Z) \ar[r]_{Hom(Id_{V} ,s_{3})} \ar[d]_{a_{1}} & Hom(U, Z ⊗ V) \\ Hom(U ⊗ V, V ⊗ Z ⊗ V) \ar[d]_{Hom(s_{5},Id_{V ⊗Z⊗V} )} & Hom(V, V ⊗ Hom(U, Z ⊗ V)) \ar[d]^{Hom(Id_{V} ,n′)} \ar[u]_{T r_{V ;Hom(U,Z⊗V )}} \\ Hom(V ⊗ U, V ⊗ Z ⊗ V) \ar[uur]^{T r_{V ;U,Z⊗V}} & Hom(V, Hom(U, V ⊗ Z ⊗ V)) \ar[l]^{q}}

The maps n′′, n′, n_{1}, as in Definition 1.39 and Notation 1.40, are all invertible, by the finite-dimensionality of V and Lemma 1.43. The commutativity of the top block, with the switching maps, is easily checked. The commutativity of the lower right triangle is Theorem 2.52. So, the claim of (2.16) is that the lower left triangle is commutative, and this will follow from showing that the outer part of the diagram is commutative, starting with v_{0} ⊗ B_{3} ∈ V ⊗ Hom(U, Z).

For v ∈ V, u ∈ U, the following maps are equal. The first step uses the formula for the inverse of the canonical map q from Lemma 1.45.

(q^{−1}((a_{1}(n′′(v_{0} ⊗ B_{3}))) ∘ s_{5}))(v) : u ↦ ((a_{1}(n′′(v_{0} ⊗ B_{3}))) ∘ s_{5})(v ⊗ u)
= [(n′′(v_{0} ⊗ B_{3})) ⊗ Id_{V}](u ⊗ v)
= v_{0} ⊗ (B_{3}(u)) ⊗ v,
n′([Id_{V} ⊗ n_{1}](v_{0} ⊗ B_{3} ⊗ v)) : u ↦ v_{0} ⊗ ((n_{1}(B_{3} ⊗ v))(u))
(2.17)  = v_{0} ⊗ (B_{3}(u)) ⊗ v.

Denote, as in Definition 1.20,

m(v_{0} ⊗ B_{3}) : 𝕂 → V ⊗ Hom(U, Z) : 1 ↦ v_{0} ⊗ B_{3},

so that

v_{0} ⊗ B_{3} ⊗ v = ([(m(v_{0} ⊗ B_{3})) ⊗ Id_{V}] ∘ l^{−1})(v),

and from (2.17),

q^{−1}((a_{1}(n′′(v_{0} ⊗ B_{3}))) ∘ s_{5}) = n′ ∘ [Id_{V} ⊗ n_{1}] ∘ [(m(v_{0} ⊗ B_{3})) ⊗ Id_{V}] ∘ l^{−1}.

The conclusion uses Corollary 2.87:

v_{0} ⊗ B_{3} ↦ T r_{V;Hom(U,Z⊗V)}((n′)^{−1} ∘ (q^{−1}((a_{1}(n′′(v_{0} ⊗ B_{3}))) ∘ s_{5})))
= T r_{V;Hom(U,Z⊗V)}([Id_{V} ⊗ n_{1}] ∘ [(m(v_{0} ⊗ B_{3})) ⊗ Id_{V}] ∘ l^{−1})
= (n_{1} ∘ s′_{3} ∘ (m(v_{0} ⊗ B_{3})))(1) = (n_{1} ∘ s′_{3})(v_{0} ⊗ B_{3}).

Example 2.90. Consider V = V_{1} = V_{2} and s_{4} : V_{1} ⊗ (V_{2} ⊗ Z) → (V_{2} ⊗ Z) ⊗ V_{1} as in the diagram from the Proof of Corollary 2.87. Then

T r_{V;V ⊗Z,Z⊗V} (s_{4}) = s_{3}.

This is a special case of Corollary 2.89, with U = V ⊗ Z, W = Z ⊗ V, s_{5} = s_{4}, B_{1} = Id_{V ⊗Z}, and B_{2} = Id_{Z⊗V}.

Example 2.91. Using Theorem 2.30 and Example 2.90,

T r_{V;Z⊗V,V ⊗Z}(s_{4}^{−1}) = T r_{V;Z⊗V,V ⊗Z}([Id_{V} ⊗ s_{3}^{−1}] ∘ s_{4} ∘ [Id_{V} ⊗ s_{3}^{−1}])
= s_{3}^{−1} ∘ (T r_{V;V ⊗Z,Z⊗V} (s_{4})) ∘ s_{3}^{−1}
= s_{3}^{−1}.

Example 2.92. Formula (2.13) from Example 2.80 also follows as a special case; for the switching map s_{1} : V ⊗V → V ⊗V as in Theorem 2.86, T r_{V;V,V} (s_{1}) = Id_{V}. This is the case of Corollary 2.89 with U = V, Z = 𝕂, B_{1} = l_{V}^{−1}, B_{2} = l, A = s_{1} = s_{5} = [Id_{V} ⊗ l] ∘ [l_{V}^{−1} ⊗ Id_{V}] ∘ s_{5}, and s_{3} : V ⊗ 𝕂 → 𝕂 ⊗ V, so

T r_{V;V,V} (s_{5}) = l ∘ s_{3} ∘ l_{V}^{−1} = Id_{V} .

Theorem 2.93. For finite-dimensional V and U, a switching map s : W ⊗U → U ⊗ W, and A : V ⊗ U → V ⊗ W,

T r_{V;U,W} (A) = T r_{V ⊗U;U,W} ([Id_{V} ⊗ s] ∘ [A ⊗ Id_{U}]).

Proof. Using Theorem 2.38, Theorem 2.30, Corollary 2.42, an easily checked equality relating the switching maps s and s′ : U ⊗ U → U ⊗ U, Theorem 2.30 again, and finally Example 2.80,

RHS = T r_{U;U,W} (T r_{V;U⊗U,U⊗W} ([Id_{V} ⊗ s] ∘ [A ⊗ Id_{U}]))
= T r_{U;U,W} (s ∘ (T r_{V;U⊗U,W ⊗U}([A ⊗ Id_{U}])))
= T r_{U;U,W} (s ∘ [(T r_{V;U,W} (A)) ⊗ Id_{U}])
= T r_{U;U,W} ([Id_{U} ⊗ (T r_{V;U,W} (A))] ∘ s′)
= (T r_{V;U,W} (A)) ∘ (T r_{U;U,U}(s′))
= T r_{V;U,W} (A).

Exercise 2.94. For finite-dimensional V and W, a switching map s : W ⊗U → U ⊗ W, and A : V ⊗ U → V ⊗ W,

T r_{V;U,W} (A) = T r_{V ⊗W;U,W} ([A ⊗ Id_{W}] ⊗ [Id_{V} ⊗ s]).

Hint. The steps are analogous to the steps in the Proof of Theorem 2.93. If U and W are both finite-dimensional, then the equality of the RHS of this equation with the RHS from Theorem 2.93 follows directly from Theorem 2.29.

2.4.2. Coevaluation and dualizability.

Theorem 2.95. For finite-dimensional V, η_{V} : 𝕂 → V ⊗ V^{∗} as in Notation 2.67, and scalar multiplication maps l_{V} : 𝕂 ⊗ V → V, l_{V^{∗}} : 𝕂 ⊗ V^{∗} → V^{∗}, l_{1} : V ⊗ 𝕂 → V, l_{2} : V^{∗} ⊗ 𝕂 → V^{∗},

l_{1} ∘ [Id_{V} ⊗ Ev_{V}] ∘ [η_{V} ⊗ Id_{V}] ∘ l_{V}^{−1} = Id_{V} ,

and

l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V^{∗}}] ∘ [Id_{V^{∗}} ⊗ η_{V}] ∘ l_{2}^{−1} = Id_{V^{∗}}.

Proof. In the following two diagrams, V = V_{1} = V_{2} = V_{3} — the subscripts are added just to track the action of the switchings and other canonical maps. In the first diagram, the upper left square uses the formula (2.9) with k^{−1} from Notation 2.67, and is commutative by Lemma 1.36. The s_{5} notation in the right half is from Corollary 2.89 and Example 2.92. The first claim is that the lower left part of the diagram is commutative.

\xymatrix{Hom(V_{2}, V_{1}) ⊗ V_{3} & V_{2}^{∗} ⊗ V_{1} ⊗ V_{3} \ar[l]_{[k⊗Id_{V} ]} \ar[d]^{[s⊗Id_{V} ]} \ar[r]^{[Id_{V ^{∗}} ⊗s_{5}]} & V_{2}^{∗} ⊗ V_{3} ⊗ V_{1} \ar[d]_{[Ev_{V} ⊗Id_{V} ]} \\ 𝕂 ⊗ V_{3} \ar[u]_{[Q_{1}^{1}⊗Id_{V} ]} \ar[r]^{[η_{V} ⊗Id_{V} ]} \ar[ddr]^{l_{V}} & V_{1} ⊗ V_{2}^{∗} ⊗ V_{3} \ar[d]^{[Id_{V} ⊗Ev_{V} ]} & 𝕂 ⊗ V_{1} \ar[ddl]_{l_{V}} \\  & V_{1} ⊗ 𝕂 \ar[d]^{l_{1}} \\  & V}

The commutativity of the right half of the diagram is easy to check. The first claim follows from checking that the identity map is equal to the composite of maps starting at V and going clockwise. Lemma 2.79 applies.

l_{1} ∘ [Id_{V} ⊗ Ev_{V}] ∘ [η_{V} ⊗ Id_{V}] ∘ l_{V}^{−1}
(2.18) = l_{V} ∘ [Ev_{V} ⊗ Id_{V}] ∘ [Id_{V^{∗}} ⊗ s_{5}] ∘ [k^{−1} ⊗ Id_{V}] ∘ [Q_{1}^{1} ⊗ Id_{V}] ∘ l_{V}^{−1}
= Id_{V} .

The expression (2.18) is also equal to T r_{V;V,V} (s_{5}) = Id_{V} as in Examples 2.80 and 2.92.

For the second claim, consider the second diagram, where s′′ is the switching involution from Lemma 2.82 and s′′′′ is another switching map so that the upper block is easily seen to be commutative.

\xymatrix{ & Hom(V_{2}, V_{1}) ⊗ V_{3}^{∗} & V_{2}^{∗} ⊗ V_{1} ⊗ V_{3}^{∗} \ar[l]^{[k⊗Id_{V ^{∗}}]} \ar[ddl]^{s′′} \\ V_{3}^{∗} ⊗ Hom(V_{2}, V_{1}) \ar[ur]^{s′′′′} & V_{3}^{∗} ⊗ V_{2}^{∗} ⊗ V_{1} \ar[l]_{[Id_{V ^{∗}} ⊗k]} \ar[d]_{[Id_{V ^{∗}} ⊗s]} \\ V_{3}^{∗} ⊗ 𝕂 \ar[u]_{[Id_{V ^{∗}} ⊗Q_{1}^{1}]} \ar[r]^{[Id_{V ^{∗}} ⊗η_{V} ]} \ar[ddr]^{l_{2}} & V_{3}^{∗} ⊗ V_{1} ⊗ V_{2}^{∗} \ar[d]_{[Ev_{V} ⊗Id_{V ^{∗}}]} \\  & 𝕂 ⊗ V_{2}^{∗} \ar[d]^{l_{V ^{∗}}} \\  & V^{∗}}

As in the first diagram, the definition of η_{V} is used in the left square, and the second claim is that the lower left part of the second diagram is commutative. The calculation is again to check the clockwise composition, and Lemma 2.82 applies.

l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V^{∗}}] ∘ [Id_{V^{∗}} ⊗ η_{V}] ∘ l_{2}^{−1}
= l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V^{∗}}] ∘ s′′ ∘ [k^{−1} ⊗ Id_{V^{∗}}] ∘ s′′′′ ∘ [Id_{V^{∗}} ⊗ Q_{1}^{1}] ∘ l_{2}^{−1}
= l_{V^{∗}} ∘ [Ev_{V} ⊗ Id_{V^{∗}}] ∘ s′′ ∘ [k^{−1} ⊗ Id_{V^{∗}}] ∘ [Q_{1}^{1} ⊗ Id_{V^{∗}}] ∘ l_{V^{∗}}^{−1}
= Id_{V^{∗}}.

Definition 2.96. A vector space V is \underline{dualizable} means: there exists (D, ϵ, η), where D is a vector space, and ϵ : D ⊗ V → 𝕂 and η : 𝕂 → V ⊗ D are linear maps such that the following diagrams (involving various scalar multiplication maps) are commutative.

\xymatrix{𝕂 ⊗ V \ar[d] \ar[r]^{[η⊗Id_{V} ]} & V ⊗ D ⊗ V \ar[d]^{[Id_{V} ⊗ϵ]} & D ⊗ 𝕂 \ar[d] \ar[r]^{[Id_{D}⊗η]} & D ⊗ V ⊗ D \ar[d]^{[ϵ⊗Id_{D}]} \\ V & V ⊗ 𝕂 \ar[l] & D & 𝕂 ⊗ D \ar[l]}

Example 2.97. Given V as in Theorem 2.95, the space D = V^{∗} and the maps ϵ = Ev_{V} and η = η_{V} = s ∘ k^{−1} ∘ Q_{1}^{1} satisfy the identities from Definition 2.96.

Remark 2.98. In category theory and other generalizations of this construction ([Stolz-Teichner], [PS]), η is called a \underline{coevaluation} map. A more general notion, with left and right duals, is considered by [Maltsiniotis].

Lemma 2.99. If V is dualizable, with duality data (D, ϵ, η), then there is an invertible map D → V^{∗}.

Proof. It is equivalent, by Example 1.28 and Lemma 1.22, to show there is an invertible map Hom(𝕂, D) → Hom(𝕂 ⊗ V, 𝕂). Denote:

(2.19)  A : Hom(𝕂, D) → Hom(𝕂 ⊗ V, 𝕂)
δ ↦ (λ ⊗ v ↦ ϵ((δ(λ)) ⊗ v)),
B : Hom(𝕂 ⊗ V, 𝕂) → Hom(𝕂, D)
ϕ ↦ (λ ↦ (l ∘ [ϕ ⊗ Id_{D}] ∘ l^{−1} ∘ η)(λ)),

where l denotes various scalar multiplications. The following diagrams are commutative, where unlabeled arrows are scalar multiplications or their inverses.

\xymatrix{𝕂 \ar[r]^{δ} \ar[d]^{η} & D \ar[d] & 𝕂 ⊗ V \ar[d]^{[η⊗Id_{V} ]} \\ V ⊗ D \ar[d] & D ⊗ 𝕂 \ar[d]^{[Id_{D}⊗η]} & V ⊗ D ⊗ V \ar[d]^{[Id_{V} ⊗ϵ]} \ar[r] & 𝕂 ⊗ V ⊗ D ⊗ V \ar[d]^{[[ϕ⊗Id_{D}]⊗Id_{V} ]} \\ 𝕂 ⊗ V ⊗ D \ar[r]^{[[δ⊗Id_{V} ]⊗Id_{D}]} \ar[dr]_{[(A(δ))⊗Id_{D}]} & D ⊗ V ⊗ D \ar[d]^{[ϵ⊗Id_{D}]} & V ⊗ 𝕂 \ar[d] & 𝕂 ⊗ D ⊗ V \ar[d] \\  & 𝕂 ⊗ D \ar[d] & V \ar[d] & D ⊗ V \ar[d]^{ϵ} \\  & D & 𝕂 ⊗ V \ar[r]^{ϕ} & 𝕂}

In the left diagram, the top square is easily checked and the lower triangle uses the formula for A. The composition in the right column gives the identity map for D by Definition 2.96, so Id_{D} ∘ δ = B(A(δ)) : 𝕂 → D.

In the right diagram, the left column gives the identity map for 𝕂 ⊗ V by Definition 2.96. For λ ⊗ v ∈ 𝕂 ⊗ V,

(A ∘ B)(ϕ) : λ ⊗ v ↦ ϵ(((B(ϕ))(λ)) ⊗ v)
= ϵ(((l ∘ [ϕ ⊗ Id_{D}] ∘ l^{−1} ∘ η)(λ)) ⊗ v)
= (ϵ ∘ [(l ∘ [ϕ ⊗ Id_{D}] ∘ l^{−1} ∘ η) ⊗ Id_{V}])(λ ⊗ v)
= (ϕ ∘ Id_{𝕂⊗V})(λ ⊗ v).

Lemma 2.100. Suppose V is dualizable, with two triples of duality data as in Definition 2.96: (D_{1}, ϵ_{1}, η_{1}) and (D_{2}, ϵ_{2}, η_{2}). Then the map a_{12} : D_{1} → D_{2},

\xymatrix{D_{1} \ar[r] & D_{1} ⊗ 𝕂 \ar[r]^{[Id_{D_{1}} ⊗η_{2}]} & D_{1} ⊗ V ⊗ D_{2} \ar[r]^{[ϵ_{1}⊗Id_{D_{2}} ]} & 𝕂 ⊗ D_{2} \ar[r] & D_{2}}

has inverse given by the map a_{21} : D_{2} → D_{1}:

\xymatrix{D_{2} \ar[r] & D_{2} ⊗ 𝕂 \ar[r]^{[Id_{D_{2}} ⊗η_{1}]} & D_{2} ⊗ V ⊗ D_{1} \ar[r]^{[ϵ_{2}⊗Id_{D_{1}} ]} & 𝕂 ⊗ D_{1} \ar[r] & D_{1}}

and a_{12} satisfies the identities [Id_{V} ⊗ a_{12}] ∘ η_{1} = η_{2} and ϵ_{2} ∘ [a_{12} ⊗ Id_{V}] = ϵ_{1}.

Proof. Some of the arrows in the following diagram are left unlabeled, but they involve only identity maps, scalar multiplications and their inverses, and the given η_{1}, η_{2}, ϵ_{1}, ϵ_{2} maps.

\xymatrix{D_{1} \ar[d] \\ D_{1} ⊗ 𝕂 \ar[dd] \ar[rr] &  & D_{1} ⊗ V ⊗ D_{2} \ar[d] \ar[dr] \\  & D_{1} ⊗ 𝕂 ⊗ V ⊗ D_{1} \ar[d] & D_{1} ⊗ V ⊗ D_{2} ⊗ 𝕂 \ar[dl] & 𝕂 ⊗ D_{2} \ar[d] \\ D_{1} ⊗ V ⊗ D_{1} \ar[ur] \ar[dd] & D_{1} ⊗ V ⊗ D_{2} ⊗ V ⊗ D_{1} \ar[d] \ar[dr] &  & D_{2} \ar[d] \\  & D_{1} ⊗ V ⊗ 𝕂 ⊗ D_{1} \ar[ul] & 𝕂 ⊗ D_{2} ⊗ V ⊗ D_{1} \ar[d] & D_{2} ⊗ 𝕂 \ar[dl] \\ 𝕂 ⊗ D_{1} \ar[d] &  & D_{2} ⊗ V ⊗ D_{1} \ar[ll] \\ D_{1}}

The composition in the left column gives the identity map D_{1} → D_{1}, and the middle left block is commutative, involving the composite [Id_{V} ⊗ ϵ_{2}] ∘ [η_{2} ⊗ Id_{V}]. The commutativity of the upper, right, and lower blocks is easy to check. The composite D_{1} → D_{2} → D_{1} clockwise from the top is equal to a_{21} ∘ a_{12}, and the commutativity of the diagram establishes the claim that a_{21} ∘ a_{12} = Id_{D_{1}}; checking the inverse in the other order follows from relabeling the subscripts.

For the identity [Id_{V} ⊗ a_{12}] ∘ η_{1} = η_{2}, consider the following diagram.

\xymatrix{𝕂 \ar[r]^{η_{2}} \ar[dd]_{η_{1}} & V ⊗ D_{2} \ar[r] & 𝕂 ⊗ V ⊗ D_{2} \ar[dd]_{[[η_{1}⊗Id_{V} ]⊗Id_{D_{2}} ]} \\  & V ⊗ 𝕂 ⊗ D_{2} \ar[u] \\ V ⊗ D_{1} \ar[r] \ar[uur]^{[Id_{V} ⊗a_{12}]} & V ⊗ D_{1} ⊗ 𝕂 \ar[r]_{[Id_{V} ⊗[Id_{D_{1}} ⊗η_{2}]]} & V ⊗ D_{1} ⊗ V ⊗ D_{2} \ar[ul]_{[Id_{V} ⊗[ϵ_{1}⊗Id_{D_{2}} ]]}}

The lower block uses the definition of a_{12}. The right block involves η_{1} and ϵ_{1} so that one of the identities from Definition 2.96 applies. The claim is that the left triangle is commutative, and this follows from the easily checked commutativity of the outer rectangle.

Similarly for the identity ϵ_{2} ∘ [a_{12} ⊗ Id_{V}] = ϵ_{1}, consider the following diagram.

\xymatrix{D_{1} ⊗ V ⊗ D_{2} ⊗ V \ar[r]^{[Id_{D_{1}} ⊗[Id_{V} ⊗ϵ_{2}]]} \ar[dd]^{[[ϵ_{1}⊗Id_{D_{2}} ]⊗Id_{V} ]} & D_{1} ⊗ V ⊗ 𝕂 \ar[r] & D_{1} ⊗ V \ar[dd]^{ϵ_{1}} \ar[ddl]^{[a_{12}⊗Id_{V} ]} \ar[dl] \\  & D_{1} ⊗ 𝕂 ⊗ V \ar[ul]^{[[Id_{D_{1}} ⊗η_{2}]⊗Id_{V} ]} \\ 𝕂 ⊗ D_{2} ⊗ V \ar[r] & D_{2} ⊗ V \ar[r]_{ϵ_{2}} & 𝕂}

The left block uses the definition of a_{12}. The top block involves η_{2} and ϵ_{2} so that one of the identities from Definition 2.96 applies. The claim is that the right triangle is commutative, and this follows from the easily checked commutativity of the outer rectangle.

In the case D = V^{∗} from Example 2.97, the maps from Lemma 2.99 and 2.100 agree (up to composition with trivial invertible maps as in the following Exercise) and so they are canonical.

Exercise 2.101. Applying Lemma 2.99 to the triple (V^{∗}, Ev_{V} , η_{V}) from Example 2.97 gives a map A such that the left diagram is commutative. If V is also dualizable with (D_{2}, ϵ_{2}, η_{2}), then the maps B from Lemma 2.99 and a_{12} from Lemma 2.100 make the right diagram commutative.

\xymatrix{V^{∗} \ar[r]^{Id_{V ^{∗}}} \ar[d]^{m} & V^{∗} \ar[d]^{Hom(l,Id_{𝕂})} & D_{2} \ar[d]^{m} & V^{∗} \ar[l]_{a_{12}} \ar[d]^{Hom(l,Id_{𝕂})} \\ Hom(𝕂, V^{∗}) \ar[r]^{A} & Hom(𝕂 ⊗ V, 𝕂) & Hom(𝕂, D_{2}) & Hom(𝕂 ⊗ V, 𝕂) \ar[l]_{B}}

Hint. The first claim is left as an exercise. For the second claim, consider ϕ ∈ V^{∗}, λ ∈ 𝕂; the following quantities agree, showing the right diagram is commutative.

(m ∘ a_{12})(ϕ) : λ ↦ (m(a_{12}(ϕ)))(λ) = λ · a_{12}(ϕ)
= λ · (l ∘ [Ev_{V} ⊗ Id_{D_{2}}] ∘ [Id_{V^{∗}} ⊗ η_{2}] ∘ l^{−1})(ϕ)
= λ · (l ∘ [Ev_{V} ⊗ Id_{D_{2}}])(ϕ ⊗ (η_{2}(1))),
(B ∘ Hom(l, Id_{𝕂}))(ϕ) : λ ↦ (B(ϕ ∘ l))(λ)
= (l ∘ [(ϕ ∘ l) ⊗ Id_{D_{2}}] ∘ l^{−1} ∘ η_{2})(λ)
= (l ∘ [(ϕ ∘ l) ⊗ Id_{D_{2}}])(1 ⊗ (η_{2}(λ)))
= (l ∘ [ϕ ⊗ Id_{D_{2}}])(η_{2}(λ)).

Lemma 2.102. If V is dualizable, with (D, ϵ, η), then D is dualizable, with duality data (V, ϵ ∘ s, s ∘ η), where s : V ⊗ D → D ⊗ V is a switching map.

Proof. In the following diagram, V = V_{1} = V_{2}.

\xymatrix{V_{1} ⊗ 𝕂 \ar[r]^{[Id_{V} ⊗η]} \ar[d] & V_{1} ⊗ V_{2} ⊗ D \ar[d]^{s_{1}} \ar[r]^{[Id_{V} ⊗s]} & V_{1} ⊗ D ⊗ V_{2} \ar[d]^{[s⊗Id_{V} ]} \\ 𝕂 ⊗ V_{1} \ar[r]^{[η⊗Id_{V} ]} \ar[d] & V_{2} ⊗ D ⊗ V_{1} \ar[d]^{[Id_{V} ⊗ϵ]} & D ⊗ V_{1} ⊗ V_{2} \ar[l]^{s_{2}} \ar[d]^{[ϵ⊗Id_{V} ]} \\ V & V_{2} ⊗ 𝕂 \ar[l] & 𝕂 ⊗ V_{2} \ar[l]}

Unlabeled arrows are obvious switching or scalar multiplication. The s_{1}, s_{2} switchings are as indicated by the subscripts. The lower left square is commutative, by the first identity from Definition 2.96, and the other small squares are easy to check, so the large square is commutative, which is the second identity for (V, ϵ ∘ s, s ∘ η) from Definition 2.96 applied to D.

Similarly, in the following diagram, D = D_{1} = D_{2}.

\xymatrix{𝕂 ⊗ D_{1} \ar[d] \ar[r]^{[η⊗Id_{D}]} & V ⊗ D_{2} ⊗ D_{1} \ar[d]^{s′_{1}} \ar[r]^{[s⊗Id_{D}]} & D_{2} ⊗ V ⊗ D_{1} \ar[d]^{[Id_{D}⊗s]} \\ D_{1} ⊗ 𝕂 \ar[d] \ar[r]^{[Id_{D}⊗η]} & D_{1} ⊗ V ⊗ D_{2} \ar[d]^{[ϵ⊗Id_{D}]} & D_{2} ⊗ D_{1} ⊗ V \ar[l]^{s′_{2}} \ar[d]^{[Id_{D}⊗ϵ]} \\ D & 𝕂 ⊗ D_{2} \ar[l] & D ⊗ 𝕂 \ar[l]}

Again, the lower left square is commutative by hypothesis, and the commutativity of the large square is the first identity for (V, ϵ∘s, s∘η) from Definition 2.96 applied to D.

Lemma 2.103. If V is dualizable, then d_{V} is invertible.

Proof. Let a_{1} : D → V^{∗} be the invertible map from Lemma 2.99, defined in terms of ϵ and A_{1} = A from (2.19). The transposes of these maps appear in the right square of the diagram.

By Lemma 2.102, D is also dualizable, with an evaluation map ϵ ∘ s, which defines A_{2} as in (2.19) and an invertible map a_{2} : V → D^{∗} from Lemma 2.99 again. These maps appear in the top square of the diagram.

\xymatrix{Hom(𝕂, V) \ar[r]^{A_{2}} & (𝕂 ⊗ D)^{∗} \\ V \ar[u]^{m_{V}} \ar[r]^{a_{2}} \ar[dr]_{d_{V}} & D^{∗} \ar[u]^{l_{D}^{∗}} & (Hom(𝕂, D))^{∗} \ar[l]^{m_{D}^{∗}} \\  & V^{∗∗} \ar[u]^{a_{1}^{∗}} & (𝕂 ⊗ V)^{∗∗} \ar[l]^{l_{V}^{∗∗}} \ar[u]_{A_{1}^{∗}}}

The two squares in the diagram are commutative by construction. The following calculation checks that a_{1}^{∗} ∘ d_{V} : V → D^{∗} is equal to a_{2}.

l_{D}^{∗} ∘ a_{1}^{∗} ∘ d_{V} = l_{D}^{∗} ∘ m_{D}^{∗} ∘ A_{1}^{∗} ∘ (l_{V}^{∗∗})^{−1} ∘ d_{V}
= ((l_{V}^{−1})^{∗} ∘ A_{1} ∘ m_{D} ∘ l_{D})^{∗} ∘ d_{V} :
v ↦ (d_{V} (v)) ∘ ((l_{V}^{−1})^{∗} ∘ A_{1} ∘ m_{D} ∘ l_{D}) :
λ ⊗ u ↦ (d_{V} (v))((A_{1}(m_{D}(λ · u))) ∘ l_{V}^{−1})
= (A_{1}(m_{D}(λ · u)))(1 ⊗ v)
= ϵ(((m_{D}(λ · u))(1)) ⊗ v)
= ϵ((λ · u) ⊗ v),
A_{2} ∘ m_{V} : v ↦ A_{2}(m_{V} (v)) :
λ ⊗ u ↦ (ϵ ∘ s)(((m_{V} (v))(λ)) ⊗ u)
= (ϵ ∘ s)((λ · v) ⊗ u)
= ϵ(u ⊗ (λ · v)).

It follows that d_{V} = (a_{1}^{∗})^{−1} ∘ a_{2} is invertible.

Theorem 2.104. Given V, the following are equivalent.

(1) k : V^{∗} ⊗ V → End(V) is invertible.

(2) V is dualizable.

(3) d : V → V^{∗∗} is invertible.

(4) V is finite-dimensional.

Proof. The Proof of Theorem 2.95 only used the property that k is invertible to show that V is dualizable, with D = V^{∗}, ϵ = Ev_{V}, and η = η_{V} = s ∘ k^{−1} ∘ Q_{1}^{1}; this is the implication (1) ⟹ (2). Lemma 2.103 just showed (2) ⟹ (3), and Theorem 2.83 showed directly that (1) ⟹ (3). The implication (3) ⟹ (4) was stated without proof in Claim 1.16 and the implication (4) ⟹ (1) was stated in Lemma 1.60, which was proved using Claim 1.34.

Remark 2.105. In the special case where (D, ϵ, η) = (V^{∗}, Ev_{V} , η_{V}) from Example 2.97, the map a_{1} from Lemma 2.103 is Id_{V^{∗}} as in Exercise 2.101, and the map a_{2} is exactly d_{V}. This shows that Lemma 2.99 (establishing that A_{2} has an inverse, B) is related to Theorem 2.83 (showing that d_{V} has an inverse); the second diagram from the Proof of Theorem 2.83 is similar to the right diagram from the Proof of Lemma 2.99.

The following result is a generalization of Theorem 2.68.

Theorem 2.106. If V is dualizable, with any triple (D_{2}, ϵ_{2}, η_{2}) as in Definition 2.96, and s_{2} : V ⊗ D_{2} → D_{2} ⊗ V is the switching map, then for any F : V ⊗ U → V ⊗ W,

(T r_{V;U,W} (F))(u)=(l_{W} ∘ [ϵ_{2} ⊗ Id_{W}] ∘ [Id_{D_{2}} ⊗ F] ∘ [(s_{2} ∘ η_{2}) ⊗ Id_{U}] ∘ l_{U}^{−1})(u).

Proof. By Theorem 2.104, V must be finite-dimensional, so the trace exists. By Theorem 2.95 (the Proof of which uses Theorem 2.68) and Example 2.97, there is another triple (D_{1}, ϵ_{1}, η_{1}) = (V^{∗}, Ev_{V} , η_{V}) satisfying Definition 2.96. There is an invertible map a_{12} : V^{∗} → D_{2} by Lemma 2.100. Consider the following diagram.

\xymatrix{ &  &  & W \\ End(V) ⊗ U & V^{∗}⊗V ⊗U \ar[l]^{[k⊗Id_{U}]} \ar[d]^{[s⊗Id_{U}]} \ar[r]^{[Id_{V ^{∗}} ⊗F]} & V^{∗}⊗V ⊗W \ar[ddr]_{[[a_{12}⊗Id_{V} ]⊗Id_{W}]} \ar[r]^{[Ev_{V} ⊗Id_{W}]} & 𝕂⊗W \ar[u]_{l_{W}} \\ 𝕂 ⊗ U \ar[u]^{[Q_{1}^{1}⊗Id_{U}]} \ar[r]^{[η_{V} ⊗Id_{U}]} \ar[dr]_{[η_{2}⊗Id_{U}]} & V ⊗V^{∗}⊗U \ar[d]^{[[Id_{V} ⊗a_{12}]⊗Id_{U}]} \\ U \ar[u]^{l_{U}^{−1}} & V ⊗ D_{2} ⊗ U \ar[r]_{[s_{2}⊗Id_{U}]} & D_{2} ⊗ V ⊗ U \ar[r]_{[Id_{D_{2}} ⊗F]} & D_{2} ⊗ V ⊗ W \ar[uu]_{[ϵ_{2}⊗Id_{W}]}}

The composition from U to W along the top row gives T r_{V;U,W} (F) by Theorem 2.68. The left square is from Theorem 2.95 and the left and right triangles are commutative by Lemma 2.100. The RHS of the Theorem is the path from U to W along the lowest row, so the claimed equality follows from the easily checked commutativity of the middle block.

Corollary 2.107. If V is dualizable, with any triple (D, ϵ, η) as in Definition 2.96, and s : V ⊗ D → D ⊗ V is the switching map, then for any A : V → V ⊗ W,

T r_{V;W} (A) = (l_{W} ∘ [ϵ ⊗ Id_{W}] ∘ [Id_{D} ⊗ A] ∘ s ∘ η)(1).

Proof. This follows from Theorem 2.106 in the same way that Corollary 2.69 follows from Theorem 2.68. By Theorem 2.54,

LHS = (T r_{V;𝕂,W} (A ∘ l_{V}))(1)
= (l_{W} ∘ [ϵ ⊗ Id_{W}] ∘ [Id_{D} ⊗ (A ∘ l_{V})] ∘ [(s ∘ η) ⊗ Id_{𝕂}] ∘ l_{𝕂}^{−1})(1)
= RHS.

This generalizes Corollary 2.69 by showing that, for any duality data (D, ϵ, η), the W -valued trace of A is the output of 1 under the composite map

\xymatrix{𝕂 \ar[r]^{η} & V ⊗ D \ar[r]^{s} & D ⊗ V \ar[r]^{[Id_{D}⊗A]} & D ⊗ V ⊗ W \ar[r]^{l_{W} ∘[ϵ⊗Id_{W}]} & W.}

Corollary 2.108. If V is dualizable, with any triple (D, ϵ, η) as in Definition 2.96, and s : V ⊗ D → D ⊗ V is the switching map, then for any A : V → V,

T r_{V} (A) = (ϵ ∘ [Id_{D} ⊗ A] ∘ s ∘ η)(1).

Proof. By Theorem 2.53 and Corollary 2.107,

LHS = T r_{V;𝕂}(l_{V}^{−1} ∘ A)
= (l_{𝕂} ∘ [ϵ ⊗ Id_{𝕂}] ∘ [Id_{D} ⊗ (l_{V}^{−1} ∘ A)] ∘ s ∘ η)(1)
= RHS.

This generalizes Theorem 2.10 and the map 𝕂 → 𝕂 from (2.6) by showing that for any (D, ϵ, η), the trace of A ∈ End(V) is the output of 1 under the composite map

\xymatrix{𝕂 \ar[r]^{η} & V ⊗ D \ar[r]^{s} & D ⊗ V \ar[r]^{[Id_{D}⊗A]} & D ⊗ V \ar[r]^{ϵ} & 𝕂.}

As mentioned at the beginning of this Section, the above results can be used as definitions of the trace: Theorem 2.106 for T r_{V;U,W}, Corollary 2.107 for T r_{V;W}, and Corollary 2.108 for T r_{V}. The proofs that these formulas are equivalent to the definitions from the previous Sections show that the trace can be calculated using any choice of duality data (D, ϵ, η), and that the output does not depend on the choice. Using this approach can also lead to simpler proofs of some properties of the trace.

Exercise 2.109. The result of Theorem 2.27,

T r_{V} (H) = (T r_{V;𝕂,𝕂}(l_{V}^{−1} ∘ H ∘ l_{V}))(1),

can be given a different (and simpler) proof using a choice of duality (D, ϵ, η).

Hint. In the following diagram,

\xymatrix{V ⊗ D ⊗ 𝕂 \ar[r]^{[s⊗Id_{𝕂}]} & D ⊗ V ⊗ 𝕂 \ar[ddr]^{[Id_{D}⊗l_{V} ]} \ar[rr]^{[Id_{D}⊗(l_{V}^{−1}∘H∘l_{V} )]} &  & D ⊗ V ⊗ 𝕂 \ar[dr]_{[ϵ⊗Id_{𝕂}]} \ar[dd]_{[Id_{D}⊗l_{V} ]} \\ 𝕂 ⊗ 𝕂 \ar[u]_{[η⊗Id_{𝕂}]} \ar[d]^{l_{1}} &  &  &  & 𝕂 ⊗ 𝕂 \ar[d]_{l_{1}} \\ 𝕂 \ar[r]^{η} & V ⊗ D \ar[r]^{s} & D ⊗ V \ar[r]^{[Id_{D}⊗H]} & D ⊗ V \ar[r]^{ϵ} & 𝕂}

the lower row is copied from Corollary 2.108, corresponding to T r_{V} (H) · Id_{𝕂} as in (2.6) from Theorem 2.27. The clockwise path from 𝕂 to 𝕂 is T r_{V;𝕂,𝕂}(l_{V}^{−1} ∘ H ∘ l_{V}), by Theorem 2.106, in the case U = W = 𝕂. The middle block in the diagram is commutative by Lemma 1.36, and the left and right blocks by versions of Lemma 1.38, and the claim follows from the commutativity of the diagram.

Remark 2.110. Theorem 2.27, Exercise 2.55, and Exercise 2.109 give some details omitted from [C_{2}] Example 2.13.

Big Exercise 2.111. Theorem 2.30, and some of the other results of Section 2.2.2, can be proved starting with Theorem 2.106 as a definition of the generalized trace.

Exercise 2.112. For any spaces D, U, V, the arrows in this diagram are invertible.

\xymatrix{Hom(U ⊗ D, Hom(V, U)) \ar[r]^{q} & Hom(U ⊗ D ⊗ V, U) \\ Hom(D ⊗ U, Hom(V, U)) \ar[u]^{Hom(s,Id_{Hom(V,U)})} & Hom(U ⊗ D ⊗ V, U ⊗ 𝕂) \ar[u]_{Hom(Id_{U⊗D⊗V} ,l_{U})}}

If V is dualizable with data (D, ϵ, η), then on the right, there is a distinguished element [Id_{U} ⊗ϵ] ∈ Hom(U ⊗D⊗V, U ⊗𝕂). In the case (D, ϵ, η) = (V^{∗}, Ev_{V} , η_{V}) from Example 2.97, there is a distinguished element k_{V U} ∈ Hom(V^{∗} ⊗ U, Hom(V, U)) on the left, and the two elements are related by the composition of arrows in the path.

Big Exercise 2.113. For a dualizable space V, define (as in [PS] §2) the \underline{mate} of a map A : V ⊗ U → V ⊗ W with respect to duality data (D, ϵ, η) as the map A^{m} : D ⊗ U → D ⊗ W given by the composition in the following diagram.

\xymatrix{D ⊗ V ⊗ U ⊗ D \ar[r]^{[Id_{D}⊗[A⊗Id_{D}]]} & D ⊗ V ⊗ W ⊗ D \ar[d]^{[ϵ⊗Id_{W ⊗D}]} \\ D ⊗ U ⊗ V ⊗ D \ar[u]^{[Id_{D}⊗[s⊗Id_{D}]]} & 𝕂 ⊗ W ⊗ D \ar[d]^{l} \\ D ⊗ U ⊗ 𝕂 \ar[u]^{[Id_{D⊗U} ⊗η]} & W ⊗ D \ar[d]^{s} \\ D ⊗ U \ar[u]^{l^{−1}} \ar[r]^{A^{m}} & D ⊗ W}

Then (as in [PS] §7),

T r_{D;U,W} (A^{m}) = T r_{V;U,W} (A).

In particular, LHS does not depend on the choice of (D, ϵ, η).

Hint. By Lemma 2.102, D is dualizable, so by Theorem 2.106, LHS exists. The formula from Theorem 2.106 does not depend on the choice of duality data for D; it is convenient to choose to use (V, ϵ ∘ s, s ∘ η) from Lemma 2.102. In the following diagram, the lower middle block uses maps from the above definition of A^{m}, including

a = [Id_{V} ⊗ [Id_{D} ⊗ [A ⊗ Id_{D}]]].
\xymatrix{V ⊗ D ⊗ U \ar[r]^{[s⊗Id_{U}]} & D ⊗ V ⊗ U \ar[r]^{[Id_{D}⊗A]} \ar[d] & D ⊗ V ⊗ W \ar[r]^{[ϵ⊗Id_{W}]} & 𝕂 ⊗ W \ar[ddd]^{l} \\ 𝕂 ⊗ U \ar[u]_{[η⊗Id_{U}]} & D ⊗ 𝕂 ⊗ V ⊗ U \ar[d] & 𝕂 ⊗ D ⊗ V ⊗ W \ar[u] \\  & D ⊗ V ⊗ D ⊗ V ⊗ U \ar[d]^{s_{0}} & D ⊗ V ⊗ D ⊗ V ⊗ W \ar[u] \\ U \ar[uu] \ar[dd] & V ⊗ D ⊗ V ⊗ U ⊗ D \ar[r]^{a} & V ⊗ D ⊗ V ⊗ W ⊗ D \ar[u] \ar[d] & W \\  & V ⊗ D ⊗ U ⊗ V ⊗ D \ar[u] & V ⊗ 𝕂 ⊗ W ⊗ D \ar[d] \\ 𝕂 ⊗ U \ar[d]^{[(s∘η)⊗Id_{U}]} & V ⊗ D ⊗ U ⊗ 𝕂 \ar[u] & V ⊗ W ⊗ D \ar[d] \\ D ⊗ V ⊗ U \ar[r]^{[s^{−1}⊗Id_{U}]} & V ⊗ D ⊗ U \ar[u] \ar[r]^{[Id_{V} ⊗A^{m}]} & V ⊗ D ⊗ W \ar[r]^{[(ϵ⊗s)⊗Id_{W}]} & 𝕂 ⊗ W \ar[uuu]_{l}}

By Theorem 2.106, the path from U to W along the top row is T r_{V;U,W} (A), and from U to W along the lowest row is T r_{D;U,W} (A^{m}). The claimed equality follows from the commutativity of the diagram. The maps in the top middle block are as in the next diagram, with notation D = D_{1} = D_{2} and V = V_{1} = V_{2} = V_{3} to indicate various switching maps.

\xymatrix{D_{2} ⊗ V_{2} ⊗ U \ar[r]^{[Id_{D}⊗A]} & D_{1} ⊗ V_{3} ⊗ W \\ D_{2} ⊗ 𝕂 ⊗ V_{2} ⊗ U \ar[u]_{[l⊗Id_{V ⊗U}]} \ar@{-->}[ur]^{[l⊗A]} \ar[d]^{[[Id_{D}⊗η]⊗Id_{V ⊗U}]} & 𝕂 ⊗ D_{1} ⊗ V_{3} ⊗ W \ar[u]^{[l⊗Id_{W}]} \\ D_{2} ⊗ V_{1} ⊗ D_{1} ⊗ V_{2} ⊗ U \ar@{-->}[ur]^{[[ϵ⊗Id_{D}]⊗A]} \ar[r]_{[Id_{D⊗V ⊗D}⊗A]} \ar[d]^{s_{0}} & D_{2} ⊗ V_{1} ⊗ D_{1} ⊗ V_{3} ⊗ W \ar[u]_{[ϵ⊗Id_{D⊗V ⊗W}]} \ar[d]^{s} \\ V_{1} ⊗ D_{1} ⊗ V_{2} ⊗ U ⊗ D_{2} \ar[r]^{a} & V_{1} ⊗ D_{1} ⊗ V_{3} ⊗ W ⊗ D_{2}}

The above diagram is commutative; the middle block involving both η and ϵ uses one of the properties from Definition 2.96. Using these maps as specified in the above right column, it is also easy to check that the right block from the big diagram is commutative.

To check that the left block from the big diagram is commutative, note that the path going up from U to D ⊗V ⊗U takes input u to output ((s∘η)(1))⊗u, and the downward path also takes u to ((s ∘ η)(1)) ⊗ u. So, it is enough to check that ((s∘η)(1))⊗u has the same output along the two paths leading to V ⊗D⊗V ⊗U ⊗D. This is shown in the next diagram, where the numbering V = V_{1} = V_{2} and D = D_{1} = D_{2} is chosen to match the left column of the previous diagram.

\xymatrix{ &  & V_{1} ⊗ D_{1} ⊗ V_{2} ⊗ U ⊗ D_{2} \ar[dl]_{[Id_{V ⊗D}⊗s]} \\ D_{2}⊗V_{1}⊗D_{1}⊗V_{2}⊗U \ar[urr]^{s_{0}} & V_{1} ⊗ D_{1} ⊗ D_{2} ⊗ V_{2} ⊗ U \ar[l]^{[s⊗Id_{V ⊗U}]} & V_{1} ⊗ D_{1} ⊗ U ⊗ V_{2} ⊗ D_{2} \ar[u]^{[Id_{V} ⊗[Id_{D}⊗[s⊗Id_{D}]]]} \ar[dl]^{[Id_{V ⊗D⊗U} ⊗s]} \\  & V_{1} ⊗ D_{1} ⊗ U ⊗ D_{2} ⊗ V_{2} \ar[u]^{[Id_{V ⊗D}⊗s_{1}^{−1}]} \\ D ⊗ 𝕂 ⊗ V ⊗ U \ar[uu]_{[[Id_{D}⊗η]⊗Id_{V ⊗U}]} \ar[dd]^{[l⊗Id_{V ⊗U}]} & 𝕂 ⊗ D ⊗ V ⊗ U \ar[l]^{[s⊗Id_{V ⊗U}]} \ar[u]^{[η⊗s_{1}]} & V ⊗ D ⊗ U ⊗ 𝕂 \ar[uu]^{[Id_{V} ⊗[Id_{D⊗V} ⊗η]]} \ar[dl]^{s} \ar[dd]_{[Id_{V} ⊗l]} \\  & 𝕂 ⊗ V ⊗ D ⊗ U \ar[dr]^{l} \ar[u]^{[[Id_{𝕂}⊗s]⊗Id_{U}]} \\ D ⊗ V ⊗ U &  & V ⊗ D ⊗ U \ar[ll]^{[s⊗Id_{U}]}}

All the blocks in the last diagram are commutative, except for the center right block. However, starting with (η(1)) ⊗ u ∈ V ⊗ D ⊗ U in the lower right corner, all paths leading upward to V ⊗ D ⊗ V ⊗ U ⊗ D give the same output. The upward right column, and the clockwise path around the left side, correspond, respectively, to the lower half and the upper half of the left block in the big diagram.

CHAPTER 3 Bilinear Forms

As a special case of Definition 1.23, consider a bilinear function V × V ⇝ 𝕂, which takes as input an ordered pair of elements of a vector space V and gives as output an element of 𝕂 (a scalar), so that it is 𝕂-linear in either input when the other is fixed. Definition 3.1 encodes this idea in a convenient way using linear maps, as described in Example 1.51. This Chapter examines the trace of a bilinear form on a finite-dimensional space V, with respect to a metric g on V.

3.1. Symmetric bilinear forms

Definition 3.1. A \underline{bilinear form} h on a vector space V is a 𝕂-linear map h : V → V^{∗}.

For vectors u, v ∈ V, a bilinear form h acts on u to give an element of the dual, h(u) ∈ V^{∗}, which then acts on v to give (h(u))(v) ∈ 𝕂.

Definition 3.2. The \underline{transpose} operator, T_{V} ∈ End(Hom(V, V^{∗})), is defined by T_{V} = Hom(d_{V} , Id_{V^{∗}}) ∘ t_{V V^{∗}} : h ↦ h^{∗} ∘ d_{V}.

Lemma 3.3. T_{V} is an involution.

Proof. The effect of the operator T_{V} is to switch the two inputs:

(3.1)  ((T_{V} (h))(u))(v) = ((h^{∗} ∘ d_{V})(u))(v) = (d_{V} (u))(h(v)) = (h(v))(u),

so the claim is obvious from Equation (3.1). This is also a corollary of Lemma 4.4 from Section 4.1, which considers some other approaches to bilinear forms and the definition of transpose, using different spaces and canonical maps.

Definition 3.4. A bilinear form h is \underline{symmetric} means: h = T_{V} (h). h is \underline{antisymmetric} means: h = −T_{V} (h).

If h is symmetric, then (h(u))(v) = (h(v))(u), and if h is antisymmetric, then (h(u))(v) = −(h(v))(u).

Notation 3.5. If \frac{1}{2} ∈ 𝕂, then the involution T_{V} on Hom(V, V^{∗}) produces, as in Lemma 1.114, a direct sum of the subspaces of symmetric and antisymmetric forms on V, denoted Hom(V, V^{∗}) = Sym(V) ⊕ Alt(V).

In particular, any bilinear form h is canonically the sum of a symmetric form and an antisymmetric form,

(3.2)  h = \frac{1}{2} · (h + T_{V} (h)) + \frac{1}{2} · (h − T_{V} (h)).

Lemma 3.6. If \frac{1}{2} ∈ 𝕂, then the canonical map k_{UU^{∗}} : U^{∗} ⊗ U^{∗} → Hom(U, U^{∗}) respects the direct sums:

k_{UU^{∗}} : S^{2}(U^{∗}) ⊕ Λ^{2}(U^{∗}) → Sym(U) ⊕ Alt(U).

Proof. The direct sums are from Example 1.118, produced by the involution s on U^{∗} ⊗U^{∗}, and Notation 3.5, produced by the involution T_{U}. It is easily checked that the following diagram is commutative.

\xymatrix{U^{∗} ⊗ U^{∗} \ar[d]^{k_{UU^{∗}}} \ar[r]^{s} & U^{∗} ⊗ U^{∗} \ar[d]^{k_{UU^{∗}}} \\ Hom(U, U^{∗}) \ar[r]^{T_{U}} & Hom(U, U^{∗})}

So k_{UU^{∗}} respects the direct sums by Lemma 1.120.

Definition 3.7. For a map H : U → V, and a bilinear form h : V → V^{∗}, the map H^{∗} ∘ h ∘ H is a bilinear form on U, called the \underline{pullback} of h.

Lemma 3.8. For any vector spaces U, V, and any map H : U → V, the following diagram is commutative.

\xymatrix{Hom(V, V^{∗}) \ar[r]^{T_{V}} \ar[d]_{Hom(H,H^{∗})} & Hom(V, V^{∗}) \ar[d]^{Hom(H,H^{∗})} \\ Hom(U, U^{∗}) \ar[r]^{T_{U}} & Hom(U, U^{∗})}

If, further, \frac{1}{2} ∈ 𝕂, then the map h ↦ H^{∗} ∘ h ∘ H respects the direct sums:

Hom(H, H^{∗}) : Sym(V) ⊕ Alt(V) → Sym(U) ⊕ Alt(U).

Proof. Using Lemma 1.6 and Lemma 1.14, the transpose of the pullback of a bilinear form h : V → V^{∗} is the pullback of the transpose:

T_{U}(H^{∗} ∘ h ∘ H) = (H^{∗} ∘ h ∘ H)^{∗} ∘ d_{U} = H^{∗} ∘ h^{∗} ∘ H^{∗∗} ∘ d_{U}
= H^{∗} ∘ h^{∗} ∘ d_{V} ∘ H = H^{∗} ∘ (T_{V} (h)) ∘ H.

The claim about the direct sums from Notation 3.5 follows from Lemma 1.120.

So, if h ∈ Sym(V), then its pullback satisfies H^{∗} ∘ h ∘ H ∈ Sym(U). The pullback of an antisymmetric form is similarly antisymmetric

Notation 3.9. If an arbitrary vector space V is a direct sum of V_{1} and V_{2}, as in Definition 1.73, and h_{1} : V_{1} → V_{1}^{∗}, h_{2} : V_{2} → V_{2}^{∗}, then

(3.3)  P_{1}^{∗} ∘ h_{1} ∘ P_{1} + P_{2}^{∗} ∘ h_{2} ∘ P_{2} : V → V^{∗}

will be called the direct sum h_{1} ⊕ h_{2} of the bilinear forms h_{1} and h_{2}.

The expression (3.3) is the same construction as in Lemma 1.82, applied to the direct sum V^{∗} = V_{1}^{∗} ⊕ V_{2}^{∗} from Example 1.80.

Theorem 3.10. T_{V} (h_{1} ⊕ h_{2}) = (T_{V_{1}}(h_{1})) ⊕ (T_{V_{2}}(h_{2})).

Proof.

LHS = (P_{1}^{∗} ∘ h_{1} ∘ P_{1})^{∗} ∘ d_{V} + (P_{2}^{∗} ∘ h_{2} ∘ P_{2})^{∗} ∘ d_{V}
= P_{1}^{∗} ∘ h_{1}^{∗} ∘ P_{1}^{∗∗} ∘ d_{V} + P_{2}^{∗} ∘ h_{2}^{∗} ∘ P_{2}^{∗∗} ∘ d_{V}
= P_{1}^{∗} ∘ h_{1}^{∗} ∘ d_{V_{1}} ∘ P_{1} + P_{2}^{∗} ∘ h_{2}^{∗} ∘ d_{V_{2}} ∘ P_{2}
= P_{1}^{∗} ∘ (T_{V_{1}}(h_{1})) ∘ P_{1} + P_{2}^{∗} ∘ (T_{V_{2}}(h_{2})) ∘ P_{2} = RHS,

using Lemma 1.6 and Lemma 1.14.

It follows that the direct sum of symmetric forms is symmetric, and that the direct sum of antisymmetric forms is antisymmetric.

The following Lemma will be convenient in some of the theorems about the tensor product of symmetric forms.

Lemma 3.11. ([B] §II.4.4) For A : U_{1} → U_{2} and B : V_{1} → V_{2}, the following diagram is commutative.

\xymatrix{U_{2}^{∗} ⊗ V_{2}^{∗} \ar[r]^{[A^{∗}⊗B^{∗}]} \ar[d]_{j} & U_{1}^{∗} ⊗ V_{1}^{∗} \ar[d]^{j} \\ Hom(U_{2} ⊗ V_{2}, 𝕂 ⊗ 𝕂) \ar[r]^{Hom([A⊗B],Id_{𝕂⊗𝕂})} \ar[d]_{Hom(Id_{U_{2}⊗V_{2}} ,l)} & Hom(U_{1} ⊗ V_{1}, 𝕂 ⊗ 𝕂) \ar[d]^{Hom(Id_{U_{1}⊗V_{1}} ,l)} \\ (U_{2} ⊗ V_{2})^{∗} \ar[r]^{[A⊗B]^{∗}} & (U_{1} ⊗ V_{1})^{∗}}

Proof. The scalar multiplication 𝕂 ⊗ 𝕂 → 𝕂 is denoted l. The top square is commutative by Lemma 1.37, and the lower one by Lemma 1.6.

Theorem 3.12. If h_{1} : U → U^{∗} and h_{2} : V → V^{∗}, then

Hom(Id_{U⊗V} , l) ∘ j ∘ [h_{1} ⊗ h_{2}] : (U ⊗ V) → (U ⊗ V)^{∗}

is a bilinear form such that

T_{U⊗V} (Hom(Id_{U⊗V} , l) ∘ j ∘ [h_{1} ⊗ h_{2}])

is equal to

Hom(Id_{U⊗V} , l) ∘ j ∘ [(T_{U}(h_{1})) ⊗ (T_{V} (h_{2}))].

Proof. First, for any U, V, the following diagram is commutative:

\xymatrix{U ⊗ V \ar[r]^{d_{U⊗V}} \ar[d]^{[d_{U} ⊗d_{V} ]} & (U ⊗ V)^{∗∗} \ar[r]^{Hom(Id_{U⊗V} ,l)^{∗}} & Hom(U ⊗ V, 𝕂 ⊗ 𝕂)^{∗} \ar[d]^{j^{∗}} \\ U^{∗∗} ⊗ V^{∗∗} \ar[r]^{j} & Hom(U^{∗} ⊗ V^{∗}, 𝕂 ⊗ 𝕂) \ar[r]^{Hom(Id_{U^{∗}⊗V ^{∗}} ,l)} & (U^{∗} ⊗ V^{∗})^{∗}}

u ⊗ v ↦ (j^{∗} ∘ Hom(Id_{U⊗V} , l)^{∗} ∘ d_{U⊗V})(u ⊗ v) :
ϕ ⊗ ξ ↦ (d_{U⊗V} (u ⊗ v))(Hom(Id_{U⊗V} , l)([ϕ ⊗ ξ]))
= l([ϕ ⊗ ξ](u ⊗ v))
= ϕ(u) · ξ(v),
u ⊗ v ↦ (Hom(Id_{U^{∗}⊗V} , l) ∘ j ∘ [d_{U} ⊗ d_{V}])(u ⊗ v)
= l ∘ [(d_{U}(u)) ⊗ (d_{V} (v))] :
ϕ ⊗ ξ ↦ ϕ(u) · ξ(v).

Note that the bottom row of the diagram is one of the columns of the diagram in Lemma 3.11 in the case U_{2} = U^{∗}, V_{2} = V^{∗}. The statement of the Theorem follows, using the above commutativity and Lemma 3.11.

T_{U⊗V} (h_{1} ⊗ h_{2}) = (Hom(Id_{U⊗V} , l) ∘ j ∘ [h_{1} ⊗ h_{2}])^{∗} ∘ d_{U⊗V}
= [h_{1} ⊗ h_{2}]^{∗} ∘ j^{∗} ∘ Hom(Id_{U⊗V} , l)^{∗} ∘ d_{U⊗V}
= [h_{1} ⊗ h_{2}]^{∗} ∘ Hom(Id_{U^{∗}⊗V^{∗}}, l) ∘ j ∘ [d_{U} ⊗ d_{V}]
= Hom(Id_{U⊗V} , l) ∘ j ∘ [h_{1}^{∗} ⊗ h_{2}^{∗}] ∘ [d_{U} ⊗ d_{V}]
= Hom(Id_{U⊗V} , l) ∘ j ∘ [(T_{U}(h_{1})) ⊗ (T_{V} (h_{2}))].

Notation 3.13. The bilinear form Hom(Id_{U⊗V} , l)∘j ∘[h_{1} ⊗h_{2}] from the above Theorem will be called the tensor product of bilinear forms, and denoted {h_{1} ⊗h_{2}}, in analogy with the bracket notation. As defined, the tensor product bilinear form acts as

({h_{1} ⊗ h_{2}}(u_{1} ⊗ v_{1}))(u_{2} ⊗ v_{2}) = (h_{1}(u_{1}))(u_{2}) · (h_{2}(v_{1}))(v_{2}).

When h_{1} and h_{2} are symmetric forms, it is clear from this formula that {h_{1}⊗h_{2}} is also symmetric, but the above proof, using Definition 3.4, makes explicit the roles of the symmetry and the scalar multiplication. It also follows that the tensor product of antisymmetric forms is symmetric.

There is a distributive law for the direct sum and tensor product of bilinear forms. Let V be a direct sum of V_{1} and V_{2}, and recall, from Example 1.77, that V ⊗ U is a direct sum of V_{1} ⊗ U and V_{2} ⊗ U, with projection maps [P_{i} ⊗ Id_{U}].

Theorem 3.14. For bilinear forms h_{1}, h_{2}, g on arbitrary vector spaces V_{1}, V_{2}, U, the following bilinear forms on V ⊗ U are equal:

{(h_{1} ⊕ h_{2}) ⊗ g} = {h_{1} ⊗ g} ⊕ {h_{2} ⊗ g}.

Proof. Unraveling the definitions, and applying Lemma 3.11 and Lemma 1.36 gives the claimed equality:

RHS = [P_{1} ⊗ Id_{U}]^{∗} ∘ Hom(Id_{V_{1}⊗U}, l) ∘ j ∘ [h_{1} ⊗ g] ∘ [P_{1} ⊗ Id_{U}]
+[P_{2} ⊗ Id_{U}]^{∗} ∘ Hom(Id_{V_{2}⊗U}, l) ∘ j ∘ [h_{2} ⊗ g] ∘ [P_{2} ⊗ Id_{U}]
= Hom(Id_{V ⊗U}, l) ∘ j ∘ [P_{1}^{∗} ⊗ Id_{U}] ∘ [h_{1} ⊗ g] ∘ [P_{1} ⊗ Id_{U}]
+Hom(Id_{V ⊗U}, l) ∘ j ∘ [P_{2}^{∗} ⊗ Id_{U}] ∘ [h_{2} ⊗ g] ∘ [P_{2} ⊗ Id_{U}]
= Hom(Id_{V ⊗U}, l) ∘ j ∘ [(P_{1}^{∗} ∘ h_{1} ∘ P_{1} + P_{2}^{∗} ∘ h_{2} ∘ P_{2}) ⊗ g]
= LHS.

3.2. Metrics

Theorem 3.15. Given a bilinear form g : V → V^{∗}, if g is symmetric or antisymmetric then the following are equivalent.

(1) V is finite-dimensional and there exists P : V^{∗} → V such that P ∘g = Id_{V}.

(2) V is finite-dimensional and there exists Q : V^{∗} → V such that g ∘ Q = Id_{V^{∗}}.

(3) g is invertible.

Proof. Let g be symmetric; the antisymmetric case is similar.

If g is invertible, then from g^{∗} ∘ d_{V} = g and Lemma 1.12,

(g^{−1})^{∗} ∘ g = (g^{∗})^{−1} ∘ g = d_{V}

is invertible, so V is finite-dimensional by Claim 1.16, which implies (1) and (2). Assuming (1) and using Claim 1.16,

Id_{V^{∗}} = Id_{V}^{∗} = (P ∘ g)^{∗} = g^{∗} ∘ P^{∗} = g ∘ d_{V}^{−1} ∘ P^{∗},

so g has a left inverse and a right inverse, and (3) follows, as in Exercise 0.48. Similarly, assuming (2),

Id_{V^{∗∗}} = Id_{V^{∗}}^{∗} = (g ∘ Q)^{∗} = Q^{∗} ∘ g^{∗} = Q^{∗} ∘ g ∘ d_{V}^{−1},

so d_{V} = Q^{∗} ∘ g ⟹ Id_{V} = d_{V}^{−1} ∘ Q^{∗} ∘ g, and g has a right inverse and a left inverse.

The invertibility condition in Theorem 3.15 implies a non-degeneracy property: for each non-zero v ∈ V, there exists a vector u ∈ V so that (g(v))(u) ≠ 0.

Definition 3.16. A \underline{metric} g on V is a symmetric, invertible map g : V → V^{∗}.

By Theorem 3.15, a metric exists only on finite-dimensional vector spaces.

Theorem 3.17. Given a metric g on V, the bilinear form d_{V} ∘g^{−1} : V^{∗} → V^{∗∗} is a metric on V^{∗}.

Proof. To show d_{V} ∘ g^{−1} is symmetric, use the definition of T_{V^{∗}} and Lemma 1.17:

T_{V^{∗}}(d_{V} ∘ g^{−1}) = (d_{V} ∘ g^{−1})^{∗} ∘ d_{V^{∗}} = (g^{−1})^{∗} ∘ d_{V}^{∗} ∘ d_{V^{∗}} = (g^{∗})^{−1} = d_{V} ∘ g^{−1}.

The last step uses the symmetry of g. This map is invertible by Theorem 3.15 and Claim 1.16.

The bilinear form d_{V} ∘ g^{−1} could be called the metric induced by g on V^{∗}, or the \underline{dual metric}. It acts on elements ϕ, ξ ∈ V^{∗} as

((d_{V} ∘ g^{−1})(ϕ))(ξ) = ξ(g^{−1}(ϕ)).

This construction can be iterated to define metrics on V^{∗∗}, etc.

Corollary 3.18. If g_{1} is a metric on V_{1} and g_{2} is a metric on V_{2}, then g_{1} ⊕g_{2} is a metric on V = V_{1} ⊕ V_{2}.

Proof. The direct sum g_{1} ⊕ g_{2} as in Notation 3.9 is symmetric by Theorem 3.10, and is invertible by Lemma 1.82. Specifically, the inclusion maps Q_{1}, Q_{2} are used to construct an inverse to the expression (3.3):

(3.4) (Q_{1} ∘ g_{1}^{−1} ∘ Q_{1}^{∗} + Q_{2} ∘ g_{2}^{−1} ∘ Q_{2}^{∗}) ∘ (P_{1}^{∗} ∘ g_{1} ∘ P_{1} + P_{2}^{∗} ∘ g_{2} ∘ P_{2}) = Id_{V} ,
(P_{1}^{∗} ∘ g_{1} ∘ P_{1} + P_{2}^{∗} ∘ g_{2} ∘ P_{2}) ∘ (Q_{1} ∘ g_{1}^{−1} ∘ Q_{1}^{∗} + Q_{2} ∘ g_{2}^{−1} ∘ Q_{2}^{∗}) = Id_{V^{∗}}.

Corollary 3.19. If g_{1} and g_{2} are metrics on U and V, then {g_{1} ⊗ g_{2}} is a metric on U ⊗ V.

Proof. The bilinear form

{g_{1} ⊗ g_{2}} = Hom(Id_{U⊗V} , l) ∘ j ∘ [g_{1} ⊗ g_{2}]

as in Notation 3.13 is symmetric by Theorem 3.12, j is invertible by the finitedimensionality (Claim 1.34), and the inverse of Hom(Id_{U⊗V} , l) ∘ j ∘ [g_{1} ⊗ g_{2}] : U ⊗ V → (U ⊗ V)^{∗} is

[g_{1}^{−1} ⊗ g_{2}^{−1}] ∘ j^{−1} ∘ Hom(Id_{U⊗V} , l^{−1}).

Exercise 3.20. If h_{1}, h_{2}, and g are metrics on V_{1}, V_{2}, and U, and h = h_{1}⊕h_{2} is the direct sum bilinear form on V = V_{1}⊕V_{2}, then the induced tensor product metric {h⊗g} on V ⊗U coincides with the induced direct sum metric on (V_{1}⊗U)⊕(V_{2}⊗U), as in Theorem 3.14.

3.3. Isometries

Example 3.21. If h is a metric on V, and H : U → V is invertible, then the pullback H^{∗} ∘ h ∘ H (as in Definition 3.7) is a metric on U, since it is symmetric by Lemma 3.8, and has inverse H^{−1} ∘ h^{−1} ∘ (H^{∗})^{−1}.

Remark 3.22. The pullback of a metric h by an arbitrary linear map H need not be a metric, for example, the case where H is the inclusion of a lightlike line in Minkowski space. See also Definition 3.106.

Definition 3.23. A 𝕂-linear map H : U → V is an \underline{isometry}, with respect to metrics g on U and h on V, means: H is invertible, and g = H^{∗} ∘ h ∘ H, so the diagram is commutative.

\xymatrix{U \ar[r]^{H} \ar[d]^{g} & V \ar[d]^{h} \\ U^{∗} & V^{∗} \ar[l]^{H^{∗}}}

This means that the metric g is equal to the pullback of h by H, and that for elements of U,

(g(u_{1}))(u_{2}) = (h(H(u_{1})))(H(u_{2})).

It follows immediately from the definition that the composite of isometries is an isometry, that the inverse of an isometry is an isometry, and that Id_{V} and −Id_{V} are isometries.

Remark 3.24. The equation g = H^{∗} ∘ h ∘ H does not itself require that H^{−1} exists, and one could consider non-surjective “isometric embeddings,” but invertibility will be assumed as part of Definition 3.23, just for convenience.

Exercise 3.25. If h : V → V^{∗} and H : U → V, and H^{∗} ∘ h ∘ H : U → U^{∗} is invertible, then H is a linear monomorphism.

Theorem 3.26. Any metric g : U → U^{∗} is an isometry with respect to itself, g, and the dual metric, d_{U} ∘ g^{−1}.

Proof. The pullback by g of the dual metric is

g^{∗} ∘ d_{U} ∘ g^{−1} ∘ g = g,

by the symmetry of g.

Theorem 3.27. Given a metric g on U, d_{U} : U → U^{∗∗} is an isometry with respect to g and the dual of the dual metric d_{U^{∗}} ∘ (d_{U} ∘ g^{−1})^{−1} = d_{U^{∗}} ∘ g ∘ d_{U}^{−1} on U^{∗∗}.

Proof. By the identity d_{U}^{∗} ∘ d_{U^{∗}} = Id_{U^{∗}} from Lemma 1.17,
g = d_{U}^{∗} ∘ d_{U^{∗}} ∘ g ∘ d_{U}^{−1} ∘ d_{U}.

Theorem 3.28. Given metrics g_{1}, g_{2}, h_{1}, and h_{2} on U_{1}, U_{2}, V_{1}, and V_{2}, if A : U_{1} → U_{2} and B : V_{1} → V_{2} are isometries, then [A ⊗ B] : U_{1} ⊗ V_{1} → U_{2} ⊗ V_{2} is an isometry with respect to the induced metrics.

Proof. The statement of the Theorem is that

{g_{1} ⊗ h_{1}} = [A ⊗ B]^{∗} ∘ {g_{2} ⊗ h_{2}} ∘ [A ⊗ B].

The RHS can be expanded, and then Lemma 3.11 applies:

RHS = [A ⊗ B]^{∗} ∘ Hom(Id_{U_{2}⊗V_{2}}, l) ∘ j ∘ [g_{2} ⊗ h_{2}] ∘ [A ⊗ B]
= Hom(Id_{U_{1}⊗V_{1}}, l) ∘ j ∘ [A^{∗} ⊗ B^{∗}] ∘ [g_{2} ⊗ h_{2}] ∘ [A ⊗ B]
= Hom(Id_{U_{1}⊗V_{1}}, l) ∘ j ∘ [g_{1} ⊗ h_{1}] = LHS.

The last step uses Lemma 1.36 and g_{1} = A^{∗} ∘ g_{2} ∘ A, h_{1} = B^{∗} ∘ h_{2} ∘ B.

Lemma 3.29. If V_{1} is finite-dimensional, then for any maps F : V_{1} → V_{2}^{∗}, E : U_{1}^{∗} → U_{2}, the following diagram is commutative.

\xymatrix{U_{1}^{∗} ⊗ V_{1} \ar[r]^{p_{U_{1}V_{1}}} \ar[d]^{[(d_{U_{2}} ∘E)⊗F]} & V_{1}^{∗∗} ⊗ U_{1}^{∗} \ar[d]^{[(F ∘d_{V_{1}}^{−1})⊗E]} \\ U_{2}^{∗∗} ⊗ V_{2}^{∗} & V_{2}^{∗} ⊗ U_{2} \ar[l]^{p_{V_{2}U_{2}}}}

Proof. The p maps are as in Notation 1.68.

ϕ ⊗ v ↦ (p_{V_{2}U_{2}} ∘ [(F ∘ d_{V_{1}}^{−1}) ⊗ E] ∘ p_{U_{1}V_{1}})(ϕ ⊗ v)
= (p_{V_{2}U_{2}} ∘ [(F ∘ d_{V_{1}}^{−1}) ⊗ E])((d_{V_{1}}(v)) ⊗ ϕ)
= p_{V_{2}U_{2}}((F (v)) ⊗ (E(ϕ)))
= (d_{U_{2}}(E(ϕ))) ⊗ (F (v)) = [(d_{U_{2}} ∘ E) ⊗ F](ϕ ⊗ v).

Theorem 3.30. Given metrics g and h on U and V, the canonical map f_{UV} : U^{∗} ⊗ V → (V^{∗} ⊗ U)^{∗} is an isometry with respect to the induced metrics.

Proof. The diagram is commutative, where the compositions in the left and right columns define the induced metrics.

\xymatrix{U^{∗} ⊗ V \ar[r]^{f_{UV}} \ar[dd]_{[(d_{U} ∘g^{−1})⊗h]} \ar[ddr]_{p_{UV}} & (V^{∗} ⊗ U)^{∗} \ar[d]^{Hom(Id_{V ^{∗}⊗U} ,l^{−1})} \\  & Hom(V^{∗} ⊗ U, 𝕂 ⊗ 𝕂) \ar[u]^{Hom(Id_{V ^{∗}⊗U} ,l)} \ar[d]^{j^{−1}} \\ U^{∗∗} ⊗ V^{∗} \ar[d]_{j} & V^{∗∗} ⊗ U^{∗} \ar[u]^{j} \ar[d]^{[(h∘d_{V}^{−1})⊗g^{−1}]} \\ Hom(U^{∗} ⊗ V, 𝕂 ⊗ 𝕂) \ar[d]_{Hom(Id_{U^{∗}⊗V} ,l)} & V^{∗} ⊗ U \ar[u]^{[(d_{V} ∘h^{−1})⊗g]} \ar[d]^{d_{V ^{∗}⊗U}} \ar[dl]^{f_{V U}} \ar[ul]^{p_{V U}} \\ (U^{∗} ⊗ V)^{∗} & (V^{∗} ⊗ U)^{∗∗} \ar[l]^{f_{UV}^{∗}}}

The lower triangle is commutative by Lemma 1.67. The two blocks with f and p maps are commutative by Lemma 1.70, and the block in the middle is commutative by Lemma 3.29.

Lemma 3.31. Given metrics g and h on U and V, let U = U_{1} ⊕ U_{2}, with operators Q_{i}, P_{i}, and let V = V_{1} ⊕ V_{2}, with operators Q′_{i}, P′_{i}. Suppose that for i = 1 or 2, the bilinear form (Q′_{i})^{∗} ∘ h ∘ Q′_{i} is a metric on V_{i}. If H : U → V is an isometry that respects the direct sums, then the bilinear form Q_{i}^{∗} ∘ g ∘ Q_{i} is a metric on U_{i}, and the induced map P′_{i} ∘ H ∘ Q_{i} : U_{i} → V_{i} is an isometry.

Proof. The induced map P′_{i} ∘ H ∘ Q_{i} is invertible, as in Lemma 1.85. The following calculation (which uses the property that H respects the direct sums) shows that the bilinear form Q_{i}^{∗} ∘ g ∘ Q_{i} is equal to the pullback of (Q′_{i})^{∗} ∘ h ∘ Q′_{i} by the map P′_{i} ∘ H ∘ Q_{i}, so it is a metric as in Example 3.21, and P′_{i} ∘ H ∘ Q_{i} is an isometry, by Definition 3.23.

(P′_{i} ∘ H ∘ Q_{i})^{∗} ∘ ((Q′_{i})^{∗} ∘ h ∘ Q′_{i}) ∘ (P′_{i} ∘ H ∘ Q_{i})
= Q_{i}^{∗} ∘ H^{∗} ∘ (P′_{i})^{∗} ∘ (Q′_{i})^{∗} ∘ h ∘ Q′_{i} ∘ P′_{i} ∘ H ∘ Q_{i}
= (Q′_{i} ∘ P′_{i} ∘ H ∘ Q_{i})^{∗} ∘ h ∘ Q′_{i} ∘ P′_{i} ∘ H ∘ Q_{i}
= (H ∘ Q_{i} ∘ P_{i} ∘ Q_{i})^{∗} ∘ h ∘ H ∘ Q_{i} ∘ P_{i} ∘ Q_{i}
= Q_{i}^{∗} ∘ H^{∗} ∘ h ∘ H ∘ Q_{i}
= Q_{i}^{∗} ∘ g ∘ Q_{i}.

3.4. Trace with respect to a metric

Definition 3.32. With respect to a metric g on V, the trace of a bilinear form h on V is defined by

T r_{g}(h) = T r_{V} (g^{−1} ∘ h).

By Lemma 2.6, this is the same as T r_{V^{∗}}(h ∘ g^{−1}), and another way to write the definition is

T r_{g} = Hom(Id_{V} , g^{−1})^{∗}(T r_{V}) ∈ Hom(V, V^{∗})^{∗}.

Theorem 3.33. Given a metric g on V, if h is any bilinear form on V, then T r_{g}(T_{V} (h)) = T r_{g}(h).

Proof.

T r_{g}(h^{∗} ∘ d_{V}) = T r_{V^{∗}}(h^{∗} ∘ d_{V} ∘ g^{−1}) = T r_{V^{∗}}(h^{∗} ∘ (g^{−1})^{∗})
= T r_{V^{∗}}((g^{−1} ∘ h)^{∗}) = T r_{V} (g^{−1} ∘ h) = T r_{g}(h),

using the symmetry of g and Lemma 2.5.

Corollary 3.34. If \frac{1}{2} ∈ 𝕂, then the trace of an antisymmetric form is 0 with respect to any metric g.

Theorem 3.35. Given a metric g on V, if T r_{V} (Id_{V}) ≠ 0, then Hom(V, V^{∗}) = 𝕂 ⊕ ker(T r_{g}).

Proof. Since T r_{g}(g) = T r_{V} (Id_{V}) ≠ 0, Lemmas 1.96 and 1.97 apply. For any h : V → V^{∗} there is a canonical decomposition of h into two terms: one that is a scalar multiple of g and the other that has trace zero with respect to g:

h = \frac{T r_{g}(h)}{T r_{V} (Id_{V})} · g + (h − \frac{T r_{g}(h)}{T r_{V} (Id_{V})} · g) .

Corollary 3.36. Given a metric g on V, if both \frac{1}{2} ∈ 𝕂 and T r_{V} (Id_{V}) ≠ 0, then Hom(V, V^{∗}) admits a direct sum 𝕂 ⊕ Sym_{0}(V, g) ⊕ Alt(V), where Sym_{0}(V, g) is the kernel of the restriction of T r_{g} to Sym(V).

Proof. Using Theorem 3.33, Theorem 1.119 applies. The canonical decomposition of any bilinear form h into three terms, corresponding to (1.11) with w = v = g, is:

h = \frac{T r_{g}(h)}{T r_{V} (Id_{V})} · g + (\frac{1}{2}(h + T_{V} (h)) − \frac{T r_{g}(h)}{T r_{V} (Id_{V})} · g) + \frac{1}{2}(h − T_{V} (h)).

Proposition 3.37. Given a metric g on V, the trace is “invariant under pullback,” that is, for an invertible map H : U → V,

T r_{H^{∗}∘g∘H}(H^{∗} ∘ h ∘ H) = T r_{g}(h).

Proof.

T r_{H^{∗}∘g∘H}(H^{∗} ∘ h ∘ H) = T r_{U}(H^{−1} ∘ g^{−1} ∘ (H^{∗})^{−1} ∘ H^{∗} ∘ h ∘ H)
= T r_{U}(H^{−1} ∘ g^{−1} ∘ h ∘ H)
= T r_{V} (g^{−1} ∘ h) = T r_{g}(h),

by Lemma 2.6.

Proposition 3.38. Given metrics g_{1}, g_{2} on V_{1}, V_{2}, if V = V_{1} ⊕ V_{2}, then for any bilinear forms h_{1} : V_{1} → V_{1}^{∗}, h_{2} : V_{2} → V_{2}^{∗},

T r_{g_{1}⊕g_{2}}(h_{1} ⊕ h_{2}) = T r_{g_{1}}(h_{1}) + T r_{g_{2}}(h_{2}).

Proof. Using the formula (3.4) for (g_{1}⊕g_{2})^{−1} from Corollary 3.18, and Lemma 2.6,

LHS = T r_{V} ((Q_{1} ∘g_{1}^{−1}∘ Q_{1}^{∗} + Q_{2} ∘ g_{2}^{−1}∘ Q_{2}^{∗}) ∘ (P_{1}^{∗} ∘ h_{1}∘ P_{1} + P_{2}^{∗} ∘ h_{2} ∘ P_{2}))
= T r_{V} (Q_{1} ∘ g_{1}^{−1} ∘ h_{1} ∘ P_{1} + Q_{2} ∘ g_{2}^{−1} ∘ h_{2} ∘ P_{2})
= T r_{V_{1}}(P_{1} ∘ Q_{1} ∘ g_{1}^{−1} ∘ h_{1}) + T r_{V_{2}}(P_{2} ∘ Q_{2} ∘ g_{2}^{−1} ∘ h_{2}) = RHS.

Proposition 3.39. Given metrics g and h on U and V, for bilinear forms E : U → U^{∗} and F : V → V^{∗},

T r_{{g⊗h}}({E ⊗ F }) = T r_{g}(E) · T r_{h}(F).

Proof. Using the formula from Corollary 3.19, there is a convenient cancellation, and then Corollary 2.36 applies:

T r_{{g⊗h}}({E ⊗ F }) = T r_{U⊗V} ([g^{−1} ⊗ h^{−1}] ∘ j^{−1} ∘ Hom(Id_{U⊗V} , l^{−1})
∘Hom(Id_{U⊗V} , l) ∘ j ∘ [E ⊗ F])
= T r_{U⊗V} (j_{2}((g^{−1} ∘ E) ⊗ (h^{−1} ∘ F)))
= T r_{g}(E) · T r_{h}(F).

3.5. THE INDUCED METRIC ON Hom(U, V) 115

3.5. The induced metric on Hom(U, V)

Definition 3.40. Given metrics g and h on U and V, define a bilinear form b on Hom(U, V), acting on elements A, B : U → V as:

(b(B))(A) = T r_{V} (A ∘ g^{−1} ∘ B^{∗} ∘ h).

b can be written as a composite:

b = Hom(Id_{Hom(U,V)}, T r_{V}) ∘ t_{V U}^{V} ∘ Hom(h, g^{−1}) ∘ t_{UV} ,

using the generalized transpose t_{V U}^{V} from Definition 1.7. By Lemma 2.6, b can also be written as a trace with respect to g:

(b(B))(A) = T r_{U}(g^{−1} ∘ B^{∗} ∘ h ∘ A) = T r_{g}(B^{∗} ∘ h ∘ A).

Theorem 3.41. Given metrics g and h on U and V, the induced tensor product metric on U^{∗} ⊗ V is equal to the pullback of the bilinear form b by the canonical map k_{UV}.

Proof. The diagram is commutative, where the composition in the left column defines the induced metric (as in Theorem 3.30), and the composition in the right column defines the bilinear form b.

\xymatrix{U^{∗} ⊗ V \ar[rr]^{k_{UV}} \ar[dd]_{[(d_{U} ∘g^{−1})⊗h]} \ar[dr]^{p_{UV}} &  & Hom(U, V) \ar[d]^{t_{UV}} \\  & V^{∗∗} ⊗ U^{∗} \ar[r]^{k_{V ^{∗}U^{∗}}} \ar[d]^{[(h∘d_{V}^{−1})⊗g^{−1}]} & Hom(V^{∗}, U^{∗}) \ar[d]^{Hom(h,g^{−1})} \\ U^{∗∗} ⊗ V^{∗} \ar[d]_{j} & V^{∗} ⊗ U \ar[r]^{k_{V U}} \ar[l]^{p_{V U}} \ar[ddl]^{f_{V U}} & Hom(V, U) \ar[d]^{t_{V U}^{V}} \ar[ddll]_{e_{V U}} \\ Hom(U^{∗}⊗ V, 𝕂⊗𝕂) \ar[d]_{Hom(Id_{U^{∗}⊗V} ,l)} &  & Hom(Hom(U, V), End(V)) \ar[d]^{Hom(Id_{Hom(U,V )},T r_{V} )} \\ (U^{∗} ⊗ V)^{∗} &  & Hom(U, V)^{∗} \ar[ll]^{k_{UV}^{∗}}}

The three squares in the upper half of the diagram are commutative, by Lemma 3.29, Lemma 1.71, and Lemma 1.58 (with h∘d_{V}^{−1} = h^{∗} because h is symmetric). The left triangle in the lower half is commutative by Lemma 1.70, and the middle triangle is just the definition f_{V U} = e_{V U} ∘k_{V U} from Notation 1.65. Checking the lower right triangle, starting with D ∈ Hom(V, U), uses (k_{UV} (ϕ ⊗ v)) ∘ D = k_{V V} ((D^{∗}(ϕ)) ⊗ v), which follows from Lemma 1.58:

Hom(D, Id_{V}) ∘ k_{UV} = k_{V V} ∘ [D^{∗} ⊗ Id_{V}],

and the definition of the trace (Definition 2.24):

D ↦ (k_{UV}^{∗} ∘ Hom(Id_{Hom(U,V)}, T r_{V}) ∘ t_{V U}^{V})(D)
= T r_{V} ∘ (t_{V U}^{V}(D)) ∘ k_{UV} :
ϕ ⊗ v ↦ T r_{V} ((t_{V U}^{V}(D))(k_{UV} (ϕ ⊗ v)))
= T r_{V} ((k_{UV} (ϕ ⊗ v)) ∘ D)
= ((k_{V V}^{−1} )^{∗}(Ev_{V}))(k_{V V} ((D^{∗}(ϕ)) ⊗ v))
= Ev_{V} ((ϕ ∘ D) ⊗ v)
= ϕ(D(v)) = (e_{V U}(D))(ϕ ⊗ v).

In particular, the pullback k_{UV}^{∗} ∘ b ∘ k_{UV} = {(d_{U} ∘ g^{−1}) ⊗ h} from Theorem 3.41 acts on ϕ, ψ ∈ U^{∗} and v, w ∈ V as:

ψ(g^{−1}(ϕ)) · (h(v))(w) = ({(d_{U} ∘ g^{−1}) ⊗ h}(ϕ ⊗ v))(ψ ⊗ w)
(3.5)  = (b(k_{UV} (ϕ ⊗ v)))(k_{UV} (ψ ⊗ w))
= T r_{U}(g^{−1} ∘ (k_{UV} (ϕ ⊗ v))^{∗} ∘ h ∘ (k_{UV} (ψ ⊗ w))).

Corollary 3.42. Given metrics g and h on U and V, b is a metric on Hom(U, V).

Proof. This follows from Example 3.21, where k_{UV}^{−1} is the invertible map relating the metric on U^{∗} ⊗ V to the bilinear form b, proving that b is symmetric and invertible, and k_{UV} and k_{UV}^{−1} are isometries.

Corollary 3.43. Given metrics g and h on U and V, the canonical map e_{UV} : Hom(U, V) → (V^{∗} ⊗ U)^{∗} is an isometry with respect to the induced metrics.

Proof. This follows from Theorem 3.30, since e_{UV} = f_{UV} ∘ k_{UV}^{−1} .

Remark 3.44. Historically, the b metric involving the trace has been called the “Hilbert-Schmidt” metric; we will just refer to it as the metric on Hom(U, V) induced by metrics on U and V and will usually use the b notation. The relationship between the metric b and the tensor product metric seems to be well-known, although possibly not in this generality. A special case of a Hermitian version of Theorem 3.41 appears in [Bhatia] §I.4, and a positive definite version for endomorphisms in [G_{2}] §III.4. Matrix versions of Theorem 3.41 appear in [Neudecker], [L], and [HJ] §4.2.

Theorem 3.45. If A : U_{2} → U_{1} is an isometry with respect to metrics g_{2}, g_{1}, and B : V_{1} → V_{2} is an isometry with respect to metrics h_{1}, h_{2}, then Hom(A, B) : Hom(U_{1}, V_{1}) → Hom(U_{2}, V_{2}) is an isometry with respect to the induced metrics.

Proof. The hypotheses are h_{1} = B^{∗} ∘ h_{2} ∘ B, and g_{2} = A^{∗} ∘ g_{1} ∘ A. For E, F ∈ Hom(U_{1}, V_{2}), the pullback of the induced metric on Hom(U_{2}, V_{2}) is

(b(B ∘ F ∘ A))(B ∘ E ∘ A) = T r_{V_{2}}(B ∘ E ∘ A ∘ g_{2}^{−1} ∘ A^{∗} ∘ F^{∗} ∘ B^{∗} ∘ h_{2})
= T r_{V_{1}}(E ∘ g_{1}^{−1} ∘ F^{∗} ∘ B^{∗} ∘ h_{2} ∘ B)
= T r_{V_{1}}(E ∘ g_{1}^{−1} ∘ F^{∗} ∘ h_{1}).

Theorem 3.46. With respect to the b metrics induced by g_{1}, g_{2}, h_{1}, h_{2} on U_{1}, U_{2}, V_{1}, V_{2}, the map j : Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) → Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) is an isometry.

Proof. For A_{1}, B_{1} : U_{1} → V_{1}, A_{2}, B_{2} : U_{2} → V_{2}, the statement of the Theorem is that the tensor product metric and pullback metric are equal:

(b(B_{1}))(A_{1}) · (b(B_{2}))(A_{2}) = (b(j(B_{1} ⊗ B_{2})))(j(A_{1} ⊗ A_{2})).

Computing the RHS, using the metrics {g_{1} ⊗ g_{2}}, and {h_{1} ⊗ h_{2}}, gives:

RHS = T r_{V_{1}⊗V_{2}}([A_{1} ⊗ A_{2}] ∘ [g_{1}^{−1} ⊗ g_{2}^{−1}] ∘ j^{−1} ∘ Hom(Id_{U_{1}⊗U_{2}}, l^{−1})
∘[B_{1} ⊗ B_{2}]^{∗} ∘ Hom(Id_{V_{1}⊗V_{2}}, l) ∘ j ∘ [h_{1} ⊗ h_{2}])
= T r_{V_{1}⊗V_{2}}([A_{1} ⊗ A_{2}] ∘ [g_{1}^{−1} ⊗ g_{2}^{−1}] ∘ [B_{1}^{∗} ⊗ B_{2}^{∗}] ∘ [h_{1} ⊗ h_{2}])
= T r_{V_{1}⊗V_{2}}([(A_{1} ∘ g_{1}^{−1} ∘ B_{1}^{∗} ∘ h_{1}) ⊗ (A_{2} ∘ g_{2}^{−1} ∘ B_{2}^{∗} ∘ h_{2})])
= T r_{V_{1}}(A_{1} ∘ g_{1}^{−1} ∘ B_{1}^{∗} ∘ h_{1}) · T r_{V_{2}}(A_{2} ∘ g_{2}^{−1} ∘ B_{2}^{∗} ∘ h_{2}).

The first step uses Lemma 3.11, and the second step uses Lemma 1.36, and finally Corollary 2.36 gives a product of traces equal to LHS.

Theorem 3.47. Given metrics g and h on U and V, the map t_{UV} : Hom(U, V) → Hom(V^{∗}, U^{∗}) is an isometry with respect to the induced metrics.

Proof. Calculating the pullback, for A, B ∈ Hom(U, V) gives

(b(B^{∗}))(A^{∗}) = T r_{U^{∗}}(A^{∗} ∘ (d_{V} ∘ h^{−1})^{−1} ∘ B^{∗∗} ∘ d_{U} ∘ g^{−1})
= T r_{U^{∗}}(A^{∗} ∘ h ∘ d_{V}^{−1} ∘ B^{∗∗} ∘ d_{U} ∘ g^{−1})
= T r_{U^{∗}}(A^{∗} ∘ h ∘ B ∘ g^{−1})
= T r_{V} (B ∘ g^{−1} ∘ A^{∗} ∘ h)
= (b(B))(A).

Corollary 3.48. Given a metric g on V, T_{V} : Hom(V, V^{∗}) → Hom(V, V^{∗}) is an isometry with respect to the induced b metric.

Proof. By Definition 3.2, T_{V} = Hom(d_{V} , Id_{V^{∗}})∘t_{V V^{∗}}, which is a composition of isometries, by Theorem 3.27, Theorem 3.45, and Theorem 3.47.

3.6. Orthogonal direct sums

Definition 3.49. A direct sum U = U_{1} ⊕ U_{2} ⊕ ⋯ ⊕ U_{ν}, with inclusion maps Q_{i}, is \underline{orthogonal} with respect to a metric g on U means: Q_{I}^{∗} ∘ g ∘ Q_{i} = 0_{Hom(U_{i},U_{I}^{∗})} for i ≠ I.

Equivalently, the direct sum is orthogonal if and only if g : U → U^{∗} respects the direct sums (as in Definition 1.84), where the direct sum structure on U^{∗} is as in Example 1.80.

Lemma 3.50. Given U with a metric g, if U = U_{1} ⊕ U_{2} and U = U′_{1} ⊕ U′_{2} are equivalent direct sums, and one is orthogonal with respect to g, then so is the other.

Proof. This follows from Lemma 1.94.

Example 3.51. Given metrics g_{1}, g_{2} on U_{1}, U_{2}, if U = U_{1} ⊕ U_{2}, then the direct sum is orthogonal with respect to the induced metric from Corollary 3.18, g = g_{1} ⊕ g_{2} = P_{1}^{∗} ∘ g_{1} ∘ P_{1} + P_{2}^{∗} ∘ g_{2} ∘ P_{2}.

Theorem 3.52. Given a metric g on U, if T r_{U}(Id_{U}) ≠ 0, then any direct sum End(U) = 𝕂⊕End_{0}(U) from Example 2.9 is orthogonal with respect to the b metric induced by g.

Proof. As noted in Lemma 1.96 and Example 2.9, any such direct sum is technically not unique, but equivalent to any other choice, so the non-uniqueness does not affect the claimed orthogonality by Lemma 3.50.

If A ∈ End_{0}(U) = ker(T r_{U}), then the b metric applied to A and any element of the line spanned by Id_{U} is T r_{U}(A ∘ g^{−1} ∘ (λ · Id_{U})^{∗} ∘ g) = λ · T r_{U}(A) = 0.

Theorem 3.53. Given a metric g on U, if T r_{U}(Id_{U}) ≠ 0, then the direct sum Hom(U, U^{∗}) = 𝕂 ⊕ ker(T r_{g}) from Theorem 3.35 is orthogonal with respect to the induced metric.

Proof. Such a direct sum is as in Lemmas 1.96 and 1.97.

If E ∈ ker(T r_{g}), then the b metric applied to E and any scalar multiple of g is T r_{U^{∗}}(E ∘ g^{−1} ∘ (λ · g)^{∗} ∘ d_{U} ∘ g^{−1}) = λ · T r_{U^{∗}}(E ∘ g^{−1}) = 0.

Lemma 3.54. Given a metric g on U, if U = U_{1} ⊕ U_{2} is an orthogonal direct sum with respect to g, then the involution on U from Example 1.116, K = Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2}, and similarly −K, are isometries with respect to g.

Proof. Using the orthogonality,

(±K)^{∗} ∘ g ∘ (±K) = (Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2})^{∗} ∘ g ∘ (Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2})
= (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2})^{∗} ∘ g ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) = g.

Lemma 3.55. Given a metric g on U, if \frac{1}{2} ∈ 𝕂 and K ∈ End(U) is an involution and an isometry with respect to g, then the direct sum produced by K, as in Lemma 1.114, is orthogonal.

Proof. Using the isometry property, K^{∗} ∘ g ∘ K = g, so using the involution property, K^{∗} ∘ g = g ∘ K. To check that g respects the direct sum, as in Definition 1.84, use Q_{i} ∘ P_{i} = \frac{1}{2} · (Id_{U} ± K) as in Lemma 1.114:

g ∘ Q_{i} ∘ P_{i} = g ∘ \frac{1}{2} · (Id_{U} ± K) = \frac{1}{2} · (Id_{U^{∗}} ± K^{∗}) ∘ g = (Q_{i} ∘ P_{i})^{∗} ∘ g = P_{i}^{∗} ∘ Q_{i}^{∗} ∘ g.

Theorem 3.56. Given a metric g on U, if \frac{1}{2} ∈ 𝕂, then the direct sum

Hom(U, U^{∗}) = Sym(U) ⊕ Alt(U)

is orthogonal with respect to the induced metric.

Proof. This follows from Lemma 3.55, Lemma 3.3, and Corollary 3.48.

Corollary 3.57. Given a metric g on U, If \frac{1}{2} ∈ 𝕂 and T r_{U}(Id_{U}) ≠ 0, then the direct sum

Hom(U, U^{∗}) = 𝕂 ⊕ Sym_{0}(U, g) ⊕ Alt(U)

is orthogonal with respect to the induced metric.

There is a converse to the construction of Example 3.51: if a direct sum is orthogonal with respect to a given metric g, then metrics are induced on the summands.

Theorem 3.58. Given a metric g on U and a direct sum U = U_{1} ⊕ U_{2} with projection and inclusion operators P_{i}, Q_{i}, if the direct sum is orthogonal with respect to g, then each of the maps g_{i} = Q_{i}^{∗} ∘ g ∘ Q_{i} : U_{i} → U_{i}^{∗} is a metric, and g = g_{1} ⊕ g_{2}.

Proof. The pullback g_{i} has inverse P_{i} ∘ g^{−1} ∘ P_{i}^{∗} by Lemma 1.85, and is symmetric by Lemma 3.8, so it is a metric. Since g respects the direct sums, P_{i}^{∗} ∘ Q_{i}^{∗} ∘ g = g ∘ Q_{i} ∘ P_{i}, so using the definition of direct sum of bilinear forms,

g_{1} ⊕ g_{2} = P_{1}^{∗} ∘ Q_{1}^{∗} ∘ g ∘ Q_{1} ∘ P_{1} + P_{2}^{∗} ∘ Q_{2}^{∗} ∘ g ∘ Q_{2} ∘ P_{2}
= g ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) = g.

Example 3.59. Theorem 3.58, applied to the above direct sums, demonstrates that under suitable hypotheses, a metric g on U induces metrics on End_{0}(U), ker(T r_{g}), Sym(U), Sym_{0}(U, g), and Alt(U).

Theorem 3.60. Given metrics g and h on U and V, if U = U_{1} ⊕ U_{2} is an orthogonal direct sum with respect to g, then the direct sum U ⊗ V = (U_{1} ⊗ V) ⊕ (U_{2}⊗V), as in Example 1.77, is orthogonal with respect to the tensor product metric {g ⊗h}, and the metric on U_{i} ⊗V induced by the direct sum coincides with {g_{i} ⊗h}.

Proof. Using Lemma 3.11, and the inclusion operators [Q_{i} ⊗ Id_{V}] : U_{i} ⊗ V → U ⊗ V,

[Q_{I} ⊗ Id_{V}]^{∗} ∘ Hom(Id_{U⊗V} , l) ∘ j ∘ [g ⊗ h] ∘ [Q_{i} ⊗ Id_{V}]
= Hom(Id_{U_{I}⊗V} , l) ∘ j ∘ [Q_{I}^{∗} ⊗ Id_{V}^{∗}] ∘ [g ⊗ h] ∘ [Q_{i} ⊗ Id_{V}]
= Hom(Id_{U_{I}⊗V} , l) ∘ j ∘ [(Q_{I}^{∗} ∘ g ∘ Q_{i}) ⊗ h].

For i ≠ I, the result is zero, showing the direct sum is orthogonal, and for i = I, the calculation shows that the tensor product of the induced metric g_{i} = Q_{i}^{∗} ∘g ∘Q_{i} and h is equal to the metric induced by {g ⊗ h} and [Q_{i} ⊗ Id_{V}] on U_{i} ⊗ V.

Theorem 3.61. Given metrics g and h on U and V, if V = V_{1} ⊕ V_{2}, with operators Q′_{i}, P′_{i}, is an orthogonal direct sum with respect to h, and U = U_{1} ⊕ U_{2}, and H : U → V is an isometry with respect to g and h which respects the direct sums, then the direct sum U_{1} ⊕ U_{2} is orthogonal with respect to g, and P′_{i} ∘ H ∘ Q_{i} : U_{i} → V_{i} is an isometry with respect to the induced metrics.

Proof. It is straightforward to check that H^{∗} : V^{∗} → U^{∗} respects the direct sums. It follows that g = H^{∗} ∘ h ∘ H is a composite of maps that respect the direct sums, so U_{1} ⊕ U_{2} is orthogonal with respect to g. The induced metrics on U_{i} and V_{i} are as in Theorem 3.58, and the last claim is a special case of Lemma 3.31.

3.7. Topics and applications

The following facts about the trace, metrics, and direct sums are left as exercises; their proofs are short and lend themselves to the methods and notation of the previous Sections. Some of the results generalize well-known properties of metrics on real vector spaces that appear in topics in geometry, algebra, or applications. A few of the results, labeled Lemmas, will be needed later.

3.7.1. Foundations of geometry.

Proposition 3.62. Let U and V be vector spaces, and let h : V → V^{∗} be an invertible 𝕂-linear map. Suppose H is just a function with domain U and target V, which is not necessarily linear, but which is right cancellable. If there is some 𝕂-linear map g : U → U^{∗} so that

((h ∘ H)(u)) ∘ H = g(u)

for all u ∈ U, then H is 𝕂-linear.

Proof. The right cancellable property in the category of sets is as in Exercise 6.18: A ∘ H = B ∘ H ⟹ A = B, for any, not necessarily linear, functions A and B.

For 𝕂-linearity, two equations must hold. First, for any λ ∈ 𝕂, u ∈ U,

((h ∘ H)(λ · u)) ∘ H = g(λ · u) = λ · g(u) = λ · ((h ∘ H)(u)) ∘ H
⟹ (h ∘ H)(λ · u) = λ · (h ∘ H)(u)
⟹ h(H(λ · u)) = h(λ · H(u))
⟹ H(λ · u) = λ · H(u).

Second, for any u_{1}, u_{2} ∈ U,

((h ∘ H)(u_{1} + u_{2})) ∘ H = g(u_{1} + u_{2}) = g(u_{1}) + g(u_{2})
= ((h ∘ H)(u_{1})) ∘ H + ((h ∘ H)(u_{2})) ∘ H
= ((h ∘ H)(u_{1}) + (h ∘ H)(u_{2})) ∘ H
⟹ (h ∘ H)(u_{1} + u_{2}) = (h ∘ H)(u_{1}) + (h ∘ H)(u_{2})
⟹ h(H(u_{1} + u_{2})) = h(H(u_{1}) + H(u_{2}))
⟹ H(u_{1} + u_{2}) = H(u_{1}) + H(u_{2}).

Proposition 3.63. Let U and V be vector spaces, and let g : U → U^{∗}, h : V → V^{∗} be symmetric bilinear forms. Suppose H is just a function with domain U and target V, which is not necessarily linear. If \frac{1}{2} ∈ 𝕂 and H(0_{U}) = 0_{V} and H satisfies

(h(H(v) − H(u)))(H(v) − H(u)) = (g(v − u))(v − u)

for all u, v ∈ U, then H also satisfies

((h ∘ H)(u)) ∘ H = g(u)

for all u ∈ U.

Proof. Expanding the RHS of the hypothesis identity using the symmetric property of g,

(g(v − u))(v − u) = (g(v))(v) − (g(v))(u) − (g(u))(v) + (g(u))(u)
= (g(v))(v) − 2(g(u))(v) + (g(u))(u).

Expanding the LHS, using the symmetric property of h and H(0_{U}) = 0_{V},

(h(H(v) − H(u)))(H(v) − H(u))
= (h(H(v)))(H(v)) − (h(H(v)))(H(u)) − (h(H(u)))(H(v)) + (h(H(u)))(H(u))
= (h(H(v) − H(0_{U})))(H(v) − H(0_{U})) − 2(h(H(u)))(H(v))
+(h(H(u) − H(0_{U})))(H(u) − H(0_{U}))
= (g(v − 0_{U}))(v − 0_{U}) − 2(h(H(u)))(H(v)) + (g(u − 0_{U}))(u − 0_{U}).

Setting the above quantities equal, cancelling like terms, and using \frac{1}{2} ∈ 𝕂, the conclusion follows.

Proposition 3.64. Let U and V be vector spaces, and let g : U → U^{∗} be a metric on U. Suppose h is just a function with domain V and target V^{∗}, which is not necessarily linear. If \frac{1}{2} ∈ 𝕂 and H : U → V is a 𝕂-linear map satisfying

((h ∘ H)(u))(H(u)) = (g(u))(u)

for all u ∈ U, then H satisfies

(h(H(v) − H(u)))(H(v) − H(u)) = (g(v − u))(v − u)

for all u, v ∈ U, and ker(H) = {0_{U}}.

Proof. To establish the claimed identity, use the linearity of H:

LHS = (h(H(v − u)))(H(v − u)) = RHS.

Suppose H(u) = 0_{V}. Then, for any v ∈ U,

((h ∘ H)(v))(H(v)) = (h(H(v) − H(u)))(H(v) − H(u))
= (g(v − u))(v − u)
= (g(v))(v) − (g(v))(u) − (g(u))(v) + (g(u))(u)
= ((h ∘ H)(v))(H(v)) − 2(g(u))(v) + ((h ∘ H)(u))(H(u)),

the last step using the symmetric property of g. Using h(H(u)) ∈ V^{∗} and H(u) = 0_{V}, the last term is 0, so cancelling like terms and using \frac{1}{2} ∈ 𝕂, the conclusion is that (g(u))(v) = 0. Since this holds for all v, g(u) = 0_{U^{∗}}, and g is invertible, so u = 0_{U}.

Corollary 3.65. Given a vector space V and metrics g and h on V, if \frac{1}{2} ∈ 𝕂 and H is just a function with domain V and target V, which is not necessarily linear, then the following are equivalent.

(1) H : V ⇝ V is right cancellable, and for all u ∈ V,

((h ∘ H)(u)) ∘ H = g(u).

(2) H : V ⇝ V is right cancellable, H(0_{V}) = 0_{V}, and for all u, v ∈ V,

(h(H(v) − H(u)))(H(v) − H(u)) = (g(v − u))(v − u).

(3) H : V → V is 𝕂-linear, and for all u ∈ V,

((h ∘ H)(u))(H(u)) = (g(u))(u).

(4) H : V → V is an isometry with respect to g and h.

Proof. For (1) ⟹ (3), the linearity is Proposition 3.62 and the identity obviously follows. Since V is finite-dimensional, a 𝕂-linear map V → V with trivial kernel must be invertible by Claim 0.50 (and therefore right cancellable), so Proposition 3.64 gives (3) ⟹ (2). (2) ⟹ (1) is Proposition 3.63. It is immediate from Definition 3.23 that (4) ⟹ (1), and also (4) ⟹ (3). Finally, the linearity of (3), the identity of (1), and the above mentioned invertibility together imply (4). The implications (1) ⟺ (4) ⟹ (3) did not require \frac{1}{2} ∈ 𝕂.

3.7.2. More isometries.

Exercise 3.66. Given metrics on U and V, the switching map s : U ⊗ V → V ⊗ U : u ⊗ v ↦ v ⊗ u, as in Example 1.29, is an isometry with respect to the induced tensor product metrics.

Lemma 3.67. Every map h : 𝕂 → 𝕂^{∗} is of the form h^{ν}, where for ν ∈ 𝕂, (h^{ν}(λ))(μ) = ν · λ · μ. If ν ≠ 0, then h^{ν} is a metric on 𝕂, with inverse map \frac{1}{ν} · T r_{𝕂}.

Proof. For any h, let ν = (h(1))(1); then (h(λ))(μ) = λ · μ · (h(1))(1). For any ν, h^{ν} = ν · h^{1}, and h^{ν} is clearly symmetric. If ν = 0 then h = 0_{Hom(𝕂,𝕂^{∗})}. If ν ≠ 0, then h^{ν} is invertible, with inverse \frac{1}{ν} · T r_{𝕂}, by Example 2.7:

((\frac{1}{ν} · T r_{𝕂}) ∘ h^{ν})(λ) = \frac{1}{ν} · T r_{𝕂}(h^{ν}(λ)) = \frac{1}{ν} · (h^{ν}(λ))(1) = \frac{1}{ν} · ν · λ · 1 = λ.

(h^{ν} ∘ (\frac{1}{ν} · T r_{𝕂}))(A) = h^{ν}(\frac{1}{ν} · A(1)) : μ ↦ ν · \frac{1}{ν} · A(1) · μ = A(μ).

A canonical such metric on 𝕂 is h^{1} = (T r_{𝕂})^{−1}. h^{1} is also equal to the map m : 𝕂 → Hom(𝕂, 𝕂) as in Definition 1.20, with inverse d_{𝕂}(1) = T r_{𝕂} as in Lemma 1.22 and Example 2.7. In particular, h^{ν} = ν · m.

Exercise 3.68. For three copies of the scalar field, 𝕂_{α}, 𝕂_{β}, 𝕂_{γ}, with metrics h^{α}, h^{β}, h^{γ}, if β = α·γ ∈ 𝕂, then the map T r_{𝕂} : Hom(𝕂_{α}, 𝕂_{β}) → 𝕂_{γ} is an isometry with respect to the induced b metric and h^{γ}.

Hint. For A, B ∈ End(𝕂), the pullback metric is, by Example 2.7,

(h^{γ}(T r_{𝕂}(A)))(T r_{𝕂}(B)) = (h^{γ}(A(1)))(B(1)) = γ · A(1) · B(1).

The induced metric b on Hom(𝕂_{α}, 𝕂_{β}) gives

T r_{𝕂}(A ∘ (h^{α})^{−1} ∘ B^{∗} ∘ h^{β}) = (A ∘ (h^{α})^{−1} ∘ B^{∗} ∘ h^{β})(1)
= A(α^{−1} · T r_{𝕂}(B^{∗}(h^{β}(1))))
= A(α^{−1} · (h^{β}(1))(B(1)))
= A(α^{−1} · β · 1 · B(1))
= \frac{β}{α} · A(1) · B(1).

Lemma 3.69. Given a metric g on U, the scalar multiplication map from Example 1.28, l_{U} : U ⊗ 𝕂 → U, is an isometry, with respect to the tensor product metric {g ⊗ h^{ν}} and ν · g.

Proof. Calculating the pullback of ν · g gives:

(ν · g(l_{U}(u_{1} ⊗ λ)))(l_{U}(u_{2} ⊗ μ)) = λ · μ · ν · (g(u_{1}))(u_{2}),

and the tensor product metric is

({g ⊗ h^{ν}}(u_{1} ⊗ λ))(u_{2} ⊗ μ) = ν · λ · μ · (g(u_{1}))(u_{2}).

Exercise 3.70. Given a metric g on U, the canonical map m : U → Hom(𝕂, U) from Definition 1.20, m(u) : λ ↦ λ · u, is an isometry with respect to g and the metric b induced by h^{1} and g.

Hint. The pullback of the metric b induced by the more general map h^{ν} is, using Lemma 3.67,

(b(m(u_{1})))(m(u_{2})) = T r_{𝕂}((h^{ν})^{−1} ∘ (m(u_{1}))^{∗} ∘ g ∘ (m(u_{1})))
= (h^{ν})^{−1}((m(u_{1}))^{∗}(g((m(u_{2}))(1))))
= ν^{−1} · T r_{𝕂}((g((m(u_{2}))(1))) ∘ (m(u_{1})))
= ν^{−1} · (g((m(u_{2}))(1)))((m(u_{1}))(1))
= ν^{−1} · (g(u_{2}))(u_{1}).

If U ≠ {0_{U}}, then ν = 1 is necessary for equality.

Lemma 3.71. Given a metric g on V, and a direct sum V = 𝕂 ⊕ U with inclusion operators Q_{i}, if the direct sum is orthogonal with respect to g, then the induced metric Q_{1}^{∗} ∘ g ∘ Q_{1} on 𝕂 is equal to h^{ν}, for ν = (g(Q_{1}(1)))(Q_{1}(1)).

Proof. Q_{1}^{∗} ∘ g ∘ Q_{1} = h^{ν} for some ν ≠ 0, by Lemma 3.67 and Theorem 3.58.

ν = (h^{ν}(1))(1) = ((Q_{1}^{∗} ∘ g ∘ Q_{1})(1))(1) = (g(Q_{1}(1)))(Q_{1}(1)).

Exercise 3.72. Given a metric g on V and an orthogonal direct sum V = 𝕂⊕U as in Lemma 3.71, if α ∈ 𝕂 satisfies

(g(Q_{1}(α)))(Q_{1}(α)) = 1,

then

d_{𝕂U}(α) : Hom(𝕂, U) → U : A ↦ A(α)

is an isometry with respect to the induced metrics.

Hint. Let h^{ν} and g_{U} be the metrics induced by g on 𝕂 and U from Lemma 3.71, so ν = (g(Q_{1}(1)))(Q_{1}(1)). For A, B ∈ Hom(𝕂, U), the pullback of g_{U} by d_{𝕂U}(α) gives:

(g_{U}((d_{𝕂U}(α))(A)))((d_{𝕂U}(α))(B)) = (g_{U}(A(α)))(B(α))
= α^{2} · (g_{U}(A(1)))(B(1)).

The calculation for the induced metric on Hom(𝕂, U) is:

(b(A))(B) = T r_{𝕂}((h^{ν})^{−1} ∘ A^{∗} ∘ g_{U} ∘ B)
= (h^{ν})^{−1}((g_{U}(B(1))) ∘ A)
= ν^{−1} · (g_{U}(B(1)))(A(1)).

If α^{2}ν = 1, then the outputs are equal; the converse holds for U ≠ {0_{U}}. The above calculation works for any metric on U, and is similar to that from Exercise 3.70.

Lemma 3.73. Given metrics on U and V, the canonical map n : Hom(U, V) ⊗ W → Hom(U, V ⊗ W) is an isometry with respect to the induced metrics.

Proof. This follows from the fact that j, l_{U}, and m are isometries, Lemma 1.42, where n = [Id_{Hom(U,V)} ⊗ m^{−1}] ∘ j ∘ Hom(l_{U}, Id_{V ⊗W}), and Theorems 3.28 and 3.45. It could also be checked directly.

Exercise 3.74. Given a metric g on U, the dual metric on U^{∗} and the metric b on Hom(U, 𝕂), induced by g and h^{1}, coincide.

Hint. For ϕ ∈ U^{∗}, and the more general metric h^{ν} on 𝕂, the identity

((h^{ν}(1)) ∘ ϕ)(λ) = ν · 1 · ϕ(λ) = (ν · ϕ)(λ)

is used in computing the b metric for ϕ, ξ ∈ U^{∗}:

T r_{𝕂}(ξ ∘ g^{−1} ∘ ϕ^{∗} ∘ h^{ν}) = ξ(g^{−1}(ϕ^{∗}(h^{ν}(1)))) = ξ(g^{−1}((h^{ν}(1)) ∘ ϕ))
= ξ(g^{−1}(ν · ϕ)) = ν · ξ(g^{−1}(ϕ)).

The dual metric on U^{∗} results in the quantity ξ(g^{−1}(ϕ)), so ν = 1 is necessary for equality in the case U ≠ {0_{U}}, and, in general, if h^{ν} is the metric on 𝕂, then the b metric on Hom(U, 𝕂) is equal to ν · d_{U} ∘ g^{−1}.

Exercise 3.75. Given metrics on U_{1} and U_{2}, if A : U_{2} → U_{1} is an isometry, then A^{∗} : U_{1}^{∗} → U_{2}^{∗} is an isometry with respect to the dual metrics.

Exercise 3.76. Given metrics on U and V, the map p : U ⊗ V → V^{∗∗} ⊗ U, as in Notation 1.68, is an isometry with respect to the induced tensor product metrics.

Exercise 3.77. Given metrics g, h, and y on U, V, and W, the map q : Hom(V, Hom(U, W)) → Hom(V ⊗ U, W), as in Definition 1.44, is an isometry with respect to the induced metrics.

Hint. All the maps in the following commutative diagram are isometries.

\xymatrix{V^{∗} ⊗ U^{∗} ⊗ W \ar[r]^{[j⊗Id_{W}]} \ar[d]^{[Id_{V ^{∗}} ⊗k_{UW}]} & Hom(V ⊗ U, 𝕂 ⊗ 𝕂) ⊗ W \ar[d]^{[Hom(Id_{V ⊗U} ,l)⊗Id_{W}]} \\ V^{∗} ⊗ Hom(U, W) \ar[d]^{k_{V,Hom(U,W)}} & (V ⊗ U)^{∗} ⊗ W \ar[d]^{k_{V ⊗U,W}} \\ Hom(V, Hom(U, W)) \ar[r]^{q} & Hom(V ⊗ U, W)}

Lemma 3.78. Given finite-dimensional vector spaces V and L, if

Ev_{L} : L^{∗} ⊗ L → 𝕂

is invertible, then d_{V L} : V → Hom(Hom(V, L), L) is invertible.

Proof. Recall d_{V L} from Definition 1.13.

\xymatrix{V \ar[r]^{d_{V L}} \ar[d]_{d_{V}} & Hom(Hom(V, L), L) \ar[d]^{Hom(k_{V L},Id_{L})} \\ V^{∗∗} \ar[dr]^{Hom(Id_{V ^{∗}} ,Q_{1}^{1})} & Hom(V^{∗} ⊗ L, L) \\ Hom(V^{∗}, L^{∗} ⊗ L) \ar[r]_{Hom(Id_{V ^{∗}} ,k_{LL})} \ar[u]^{Hom(Id_{V ^{∗}} ,Ev_{L})} & Hom(V^{∗}, End(L)) \ar[u]_{q}}

The lower triangle is commutative, by the calculation from the Proof of Proposition 2.21, and so is the upper part of the diagram:

v ↦ (q ∘ Hom(Id_{V^{∗}}, Q_{1}^{1}) ∘ d_{V})(v)
= q(Q_{1}^{1} ∘ (d_{V} (v))) :
ϕ ⊗ u ↦ (Q_{1}^{1}(ϕ(v)))(u) = (ϕ(v) · Id_{L})(u) = ϕ(v) · u,
v ↦ (Hom(k_{V L}, Id_{L}) ∘ d_{V L})(v)
= (d_{V L}(v)) ∘ k_{V L} :
ϕ ⊗ u ↦ (k_{V L}(ϕ ⊗ u))(v) = ϕ(v) · u.

d_{V L} is invertible because all the other maps in the outer rectangle are invertible.

Proposition 3.79. For V, L, and Ev_{L} as in Lemma 3.78, if there is some metric y on L, and Ev_{L} is an isometry with respect to the induced metric on L^{∗} ⊗L and h^{1} on 𝕂, then d_{V L} is an isometry with respect to any metric g on V, and the induced metric on Hom(Hom(V, L), L).

Proof. By Theorem 3.45 and the hypothesis on Ev_{L}, Hom(Id_{V^{∗}}, Ev_{L}) is an isometry with respect to the b metric on Hom(V^{∗}, L^{∗} ⊗ L), and the b metric on V^{∗∗} = Hom(V^{∗}, 𝕂), induced by d_{V} ∘g^{−1} on V^{∗} and h^{1} on 𝕂. By Theorem 3.27, d_{V} is an isometry from V to V^{∗∗} with respect to the dual metric, which by Exercise 3.74, is the same as the b metric on Hom(V^{∗}, 𝕂). Referring to the diagram from Lemma 3.78, d_{V L} is equal to a composite of isometries.

3.7.3. Antisymmetric forms and symplectic forms.

Exercise 3.80. Given a bilinear form g : V → V^{∗}, if g satisfies (g(v))(v) = 0 for all v ∈ V, then g is antisymmetric. If \frac{1}{2} ∈ 𝕂, then, conversely, an antisymmetric form g satisfies (g(v))(v) = 0.

Big Exercise 3.81. Given a bilinear form g : V → V^{∗}, the following are equivalent.

(1) For all u, v ∈ V, if (g(u))(v) = 0, then (g(v))(u) = 0.

(2) g ∈ Sym(V) ∪ Alt(V).

Hint. (2) ⟹ (1) is easy; a proof of the well-known converse is given by [J] §6.1. A bilinear form satisfying either equivalent condition is variously described by the literature as “orthosymmetric” or “reflexive.”

Definition 3.82. A bilinear form h : U → U^{∗} is a \underline{symplectic} form means: h is antisymmetric and invertible.

Recall from Theorem 3.15 that the invertibility implies U is finite-dimensional.

Exercise 3.83. Given a symplectic form h on V, the bilinear form d_{V} ∘ h^{−1} : V^{∗} → V^{∗∗} is a symplectic form on V^{∗}.

Hint. This is an analogue of Theorem 3.17. The antisymmetric property implies the equality d_{V} ∘ h^{−1} = −(h^{∗})^{−1}.

Given a symplectic form h on V, the above Exercise suggests there are two opposite ways h could induce a symplectic form on V^{∗}:

(3.6)  d_{V} ∘ h^{−1} = −(h^{∗})^{−1},
(3.7)  −d_{V} ∘ h^{−1} = (h^{∗})^{−1}.

Exercise 3.84. The tensor product of symplectic forms is a metric.

The following Definition is analogous to Definition 3.23.

Definition 3.85. A map H : U → V is a \underline{symplectic isometry}, with respect to symplectic forms g on U, and h on V, means: H is invertible, and g = H^{∗} ∘ h ∘ H.

Lemma 3.86. A symplectic form h : U → U^{∗} is a symplectic isometry with respect to itself and the symplectic form −d_{V} ∘ h^{−1} from (3.7).

Exercise 3.87. Given V with metric g and symplectic form h, the following are equivalent.

(1) g is a symplectic isometry with respect to h and the symplectic form d_{V} ∘ h^{−1} from (3.6).

(2) h^{−1} ∘ g ∈ End(V) is an involution.

Exercise 3.88. Given V with metric g and symplectic form h, the following are equivalent.

(1) g is a symplectic isometry with respect to h and the symplectic form −d_{V} ∘ h^{−1} from (3.7).

(2) h is an isometry with respect to g and the dual metric d_{V} ∘ g^{−1}.

(3) g^{−1} ∘ h ∈ End(V) is an isometry with respect to g.

(4) g^{−1} ∘ h ∈ End(V) is a symplectic isometry with respect to h.

Hint. The equivalence of (2) and (3) follows from Theorem 3.26.

Exercise 3.89. Given a symplectic form h on U, using either method (3.6) or (3.7) to induce a symplectic form on the dual space, the double dual U^{∗∗} has a canonical symplectic form

d_{U^{∗}} ∘ (d_{U} ∘ h^{−1})^{−1} = −d_{U^{∗}} ∘ (−d_{U} ∘ h^{−1})^{−1} = d_{U^{∗}} ∘ h ∘ d_{U}^{−1}.

The map d_{U} : U → U^{∗∗} is a symplectic isometry with respect to h and the above symplectic form.

Big Exercise 3.90. Several more of the elementary results on metrics can be adapted to symplectic forms.

3.7.4. More direct sums.

Exercise 3.91. Given linear maps H : U → V and h : V → V^{∗}, if H^{∗} ∘ h ∘ H : U → U^{∗} is invertible, then there is a direct sum V = U ⊕ker(H^{∗}∘h). If, in addition, h is symmetric (or antisymmetric), then h : V → V^{∗} respects the induced direct sums and H^{∗} ∘ h ∘ H : U → U^{∗} is a metric (respectively, symplectic form) on U. If, further, h is invertible, then h also induces a metric (respectively, symplectic form) on ker(H^{∗} ∘ h).

Hint. H is a linear monomorphism as in Exercise 3.25. Let Q_{1} = H, and let P_{1} = (H^{∗} ∘ h ∘ H)^{−1} ∘ H^{∗} ∘ h. Then P_{1} ∘ Q_{1} = Id_{U}, and Q_{1} ∘ P_{1} = H ∘ (H^{∗} ∘ h ∘ H)^{−1} ∘H^{∗} ∘h is an idempotent on V. The kernel of Q_{1} ∘P_{1} is equal to the kernel of H^{∗} ∘ h; let Q_{2} denote the inclusion of this subspace in V, and define the projection P_{2} onto this subspace as in Example 1.109: P_{2} = Id_{V} − Q_{1} ∘ P_{1} = Q_{2} ∘ P_{2}.

The direct sum V = U ⊕ ker(H^{∗} ∘ h) induces a direct sum V^{∗} = U^{∗} ⊕ (ker(H^{∗} ∘ h))^{∗} as in Example 1.80. If h is symmetric (or antisymmetric), then H^{∗}∘h∘H is also symmetric (respectively, antisymmetric) by Lemma 3.8, and a metric (respectively, symplectic form) on U, so U is finite-dimensional and d_{U} is invertible. Consider the two expressions:

h ∘ Q_{1} ∘ P_{1} = h ∘ H ∘ (H^{∗} ∘ h ∘ H)^{−1} ∘ H^{∗} ∘ h,
P_{1}^{∗} ∘ Q_{1}^{∗} ∘ h = h^{∗} ∘ H^{∗∗} ∘ (H^{∗} ∘ h^{∗} ∘ H^{∗∗})^{−1} ∘ H^{∗} ∘ h.

If h = ±h^{∗} ∘ d_{V}, then, using Lemma 1.14,

h ∘ Q_{1} ∘ P_{1} = ±h^{∗} ∘ d_{V} ∘ H ∘ (H^{∗} ∘ (±h^{∗} ∘ d_{V}) ∘ H)^{−1} ∘ H^{∗} ∘ h
= h^{∗} ∘ H^{∗∗} ∘ d_{U} ∘ (H^{∗} ∘ h^{∗} ∘ H^{∗∗} ∘ d_{U})^{−1} ∘ H^{∗} ∘ h,

so h ∘ Q_{1} ∘ P_{1} = P_{1}^{∗} ∘ Q_{1}^{∗} ∘ h, and h respects the direct sums.

If, further, h is invertible, then h is a metric (respectively, symplectic form) that respects the direct sums V → V^{∗}, so V = U ⊕ ker(H^{∗} ∘ h) is an orthogonal direct sum with respect to h, and Theorem 3.58 applies.

Exercise 3.92. Given V = V_{1}⊕V_{2}, U = U_{1}⊕U_{2}, with projection and inclusion maps P_{i}, Q_{i} on V, P′_{i}, Q′_{i} on U, if A : U_{1} → V_{1} and B : U_{2} → V_{2} are isometries with respect to metrics g_{i} on U_{i}, h_{i} on V_{i}, then

A ⊕ B = Q_{1} ∘ A ∘ P′_{1} + Q_{2} ∘ B ∘ P′_{2} : U → V

is an isometry with respect to the induced metrics.

Hint. The invertibility is by Lemma 1.82. The rest of the claim is that

g_{1} ⊕ g_{2} = (A ⊕ B)^{∗} ∘ (h_{1} ⊕ h_{2}) ∘ (A ⊕ B).

The RHS can be expanded:

RHS = (P_{1}^{′∗} ∘ A^{∗} ∘ Q_{1}^{∗} + P_{2}^{′∗} ∘ B^{∗} ∘ Q_{2}^{∗})
∘(P_{1}^{∗} ∘ h_{1} ∘ P_{1} + P_{2}^{∗} ∘ h_{2} ∘ P_{2})
∘(Q_{1} ∘ A ∘ P′_{1} + Q_{2} ∘ B ∘ P′_{2})
= P_{1}^{′∗} ∘ A^{∗} ∘ h_{1} ∘ A ∘ P′_{1} + P_{2}^{′∗} ∘ B^{∗} ∘ h_{2} ∘ B ∘ P′_{2}
= P_{1}^{′∗} ∘ g_{1} ∘ P′_{1} + P_{2}^{′∗} ∘ g_{2} ∘ P′_{2} = LHS.

The last step uses g_{1} = A^{∗} ∘ h_{1} ∘ A, g_{2} = B^{∗} ∘ h_{2} ∘ B.

Exercise 3.93. Given metrics g_{1} and g_{2} on V_{1} and V_{2}, if V = V_{1} ⊕ V_{2} and W = V_{1} ⊕ V_{2} are direct sums with operators P′_{i}, Q′_{i} and P_{i}, Q_{i}, respectively, then the map Q′_{1} ∘ P_{1} + Q′_{2} ∘ P_{2} : W → V is an isometry with respect to the direct sum metrics from Corollary 3.18.

Hint. This is a special case of Exercise 3.92. The construction of the invertible map Q′_{1} ∘ P_{1} + Q′_{2} ∘ P_{2} : W → V is a special case of the map from Lemma 1.82.

Exercise 3.94. Given metrics g_{1} and g_{2} on V_{1} and V_{2}, if V = V_{1} ⊕ V_{2}, then the dual of the metric g_{1} ⊕g_{2} from Corollary 3.18 is d_{V} ∘(g_{1} ⊕g_{2})^{−1} : V^{∗} → V^{∗∗}, as in Theorem 3.17. For the direct sum V^{∗} = V_{1}^{∗} ⊕ V_{2}^{∗} from Example 1.80, the direct sum of the dual metrics is (d_{V_{1}} ∘ g_{1}^{−1}) ⊕ (d_{V_{2}} ∘ g_{2}^{−1}). These two metrics on V^{∗} are equal.

Hint. Lemma 1.14 applies to the direct sum formula (3.3) and the inverse (3.4).

Example 3.95. Given \frac{1}{2} ∈ 𝕂, and given V with metric g and an involution K_{1} : V → V, producing a direct sum V_{1} ⊕ V_{2} as in Lemma 1.114, suppose the bilinear forms Q_{i}^{∗} ∘ g ∘ Q_{i} are metrics for i = 1, 2 (this is the case, for example, when K_{1} is an isometry, by Lemma 3.55 and Theorem 3.58). If K_{2} is another involution on V that is an isometry and anticommutes with K_{1}, then K_{2} respects the direct sums V_{1} ⊕ V_{2} → V_{2} ⊕ V_{1} as in Lemma 1.121, and the induced maps P_{2} ∘ K_{2} ∘ Q_{1} : V_{1} → V_{2} and P_{1} ∘ K_{2} ∘ Q_{2} : V_{2} → V_{1}, as in Theorem 1.130, are isometries by Lemma 3.31.

Lemma 3.96. Given \frac{1}{2} ∈ 𝕂, and given V with metric g and an involution K : V → V, producing a direct sum V = V_{1} ⊕ V_{2} with operators P_{i}, Q_{i} as in Lemma 1.114, suppose the direct sum is orthogonal with respect to g (this is the case, for example, when K is an isometry, by Lemma 3.55). Let K′ be another involution on V that is an isometry and anticommutes with K, and which produces a direct sum V = V′_{1} ⊕ V′_{2}, with operators P′_{i}, Q′_{i}. If β ∈ 𝕂 satisfies β^{2} = 2, then for i = 1, 2, I = 1, 2, the map β · P′_{I} ∘ Q_{i} : V_{i} → V′_{I} is an isometry.

Proof. The map β · P′_{I} ∘ Q_{i} : V_{i} → V′_{I} is invertible by Theorem 1.131. The induced metric on V_{i} is Q_{i}^{∗} ∘ g ∘ Q_{i} and on V′_{I} is (Q′_{I})^{∗} ∘ g ∘ Q′_{I}, by Lemma 3.55 and Theorem 3.58. From the Proof of Lemma 3.55, g ∘ Q′_{I} ∘ P′_{I} = (P′_{I})^{∗} ∘ (Q′_{I})^{∗} ∘ g.

(β · P′_{I} ∘ Q_{i})^{∗} ∘ ((Q′_{I})^{∗} ∘ g ∘ Q′_{I}) ∘ (β · P′_{I} ∘ Q_{i})
= β^{2} · Q_{i}^{∗} ∘ (P′_{I})^{∗} ∘ (Q′_{I})^{∗} ∘ g ∘ Q′_{I} ∘ P′_{I} ∘ Q_{i}
= β^{2} · Q_{i}^{∗} ∘ g ∘ Q′_{I} ∘ P′_{I} ∘ Q_{i}
= β^{2} · Q_{i}^{∗} ∘ g ∘ \frac{1}{2} · (Id_{V} ± K′) ∘ Q_{i}.

By hypothesis, g respects the direct sum V_{1} ⊕ V_{2}, but K′ reverses the direct sum as in Lemma 1.121. So, Q_{i}^{∗} ∘ g ∘ K′ ∘ Q_{i} = 0_{Hom(V_{i},V_{i}^{∗})} and the second term in the last line drops out.

Exercise 3.97. Given metrics g and h on U and V, let U = U_{1} ⊕ U_{2} and V = V_{1} ⊕V_{2} be orthogonal direct sums with operators P_{i}, Q_{i}, P′_{i}, Q′_{i}. If H : U → V is an isometry such that P′_{2} ∘ H ∘ Q_{1} = 0_{Hom(U_{1},V_{2})}, and P′_{1} ∘ H ∘ Q_{1} is a linear epimorphism, then P′_{1} ∘ H ∘ Q_{2} = 0_{Hom(U_{2},V_{1})}, so H respects the direct sums.

Hint.

0_{Hom(U_{1},U_{2}^{∗})} = Q_{2}^{∗} ∘ g ∘ Q_{1}
= Q_{2}^{∗} ∘ H^{∗} ∘ h ∘ H ∘ Q_{1}
= Q_{2}^{∗} ∘ H^{∗}∘ (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2})^{∗}∘h∘ (Q′_{1} ∘ P′_{1}+Q′_{2} ∘ P′_{2}) ∘ H ∘ Q_{1}
= (P′_{1} ∘ H ∘ Q_{2})^{∗} ∘ Q_{1}^{′∗} ∘ h ∘ Q′_{1} ∘ P′_{1} ∘ H ∘ Q_{1}.

Q_{1}^{′∗} ∘ h ∘ Q′_{1} is invertible by Theorem 3.58, so P′_{1} ∘ H ∘ Q_{2} = 0_{Hom(U_{2},V_{1})} by the linear epimorphism property (Definition 0.46).

Exercise 3.98. If U_{1} = V_{1} in Exercise 3.97, then the epimorphism property is not needed in the hypothesis.

Hint.

(P_{1} ∘ H^{−1} ∘ Q′_{1}) ∘ (P′_{1} ∘ H ∘ Q_{1}) = P_{1} ∘ H^{−1} ∘ (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2}) ∘ H ∘ Q_{1}
= P_{1} ∘ Q_{1} = Id_{V_{1}}.

Claim 0.50 applies.

Exercise 3.99. Given any vector space V, if U = U_{1} ⊕ U_{2} is a direct sum with projection operators P_{i} and inclusion operators Q_{i}, then as in Example 1.79, Hom(U, V) = Hom(U_{1}, V) ⊕ Hom(U_{2}, V), with projection operators Hom(Q_{i}, Id_{V}) and inclusion operators Hom(P_{i}, Id_{V}). Given metrics g and h on U and V, if U_{1}⊕U_{2} is an orthogonal direct sum, then Hom(U_{1}, V)⊕Hom(U_{2}, V) is an orthogonal direct sum with respect to the induced b metric.

Hint. Consider A : U_{i} → V, B : U_{I} → V.
((Hom(P_{I}, Id_{V})^{∗}∘ b ∘ Hom(P_{i}, Id_{V}))(A))(B) = (b(A ∘ P_{i}))(B ∘ P_{I})
= T r_{V} (B ∘ P_{I} ∘ g^{−1}∘ (A ∘ P_{i})^{∗}∘ h)
= T r_{V} (B ∘ P_{I} ∘ g^{−1}∘ P_{i}^{∗}∘ A^{∗}∘ h).

By Lemma 1.85, since g : U → U^{∗} respects the direct sums, so does g^{−1} : U^{∗} → U, so for i ≠ I, P_{I} ∘ g^{−1} ∘ P_{i}^{∗} = 0_{Hom(U_{i}^{∗},U_{I})}. This makes (b(A ∘ P_{i}))(B ∘ P_{I}) equal to zero, proving orthogonality.

Exercise 3.100. Given any vector space U, if V = V_{1} ⊕ V_{2} is a direct sum with projection operators P_{i} and inclusion operators Q_{i}, then as in Example 1.78, Hom(U, V) = Hom(U, V_{1}) ⊕ Hom(U, V_{2}), with projection operators Hom(Id_{U}, P_{i}) and inclusion operators Hom(Id_{U}, Q_{i}). Given metrics g and h on U and V, if V_{1}⊕V_{2} is an orthogonal direct sum, then Hom(U, V_{1}) ⊕ Hom(U, V_{2}) is an orthogonal direct sum with respect to the induced b metric.

Hint. Consider A : U → V_{i}, B : U → V_{I}.
((Hom(Id_{U}, Q_{I})^{∗}∘b∘Hom(Id_{U}, Q_{i}))(A))(B) = (b(Q_{i} ∘ A))(Q_{I} ∘ B)
= T r_{V} (Q_{I} ∘B∘g^{−1}∘(Q_{i}∘A)^{∗}∘h)
= T r_{V_{I}} (B∘g^{−1}∘A^{∗}∘ Q_{i}^{∗}∘h∘Q_{I}).

For i ≠ I, this quantity is zero.

Exercise 3.101. Given a metric g on U, if U = U_{1} ⊕U_{2} is an orthogonal direct sum with operators Q_{i}, P_{i}, and g_{1}, g_{2} are the metrics induced on U_{1}, U_{2} (from Theorem 3.58), and K : U → U^{∗}, then

T r_{g}(K) = T r_{g_{1}}(Q_{1}^{∗} ∘ K ∘ Q_{1}) + T r_{g_{2}}(Q_{2}^{∗} ∘ K ∘ Q_{2}).

Hint. By Theorem 3.58, g_{i}^{−1} = P_{i} ∘ g^{−1} ∘ P_{i}^{∗}, and from the hint for Exercise 3.99, P_{I} ∘ g^{−1} ∘ P_{i}^{∗} = 0_{Hom(U_{i}^{∗},U_{I})} for i ≠ I. Using Lemma 2.6,

T r_{g}(K) = T r_{V} (g^{−1} ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2})^{∗} ∘ K ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}))
= T r_{V_{1}}(P_{1} ∘ g^{−1} ∘ (P_{1}^{∗} ∘ Q_{1}^{∗} + P_{2}^{∗} ∘ Q_{2}^{∗}) ∘ K ∘ Q_{1})
+T r_{V_{2}}(P_{2} ∘ g^{−1} ∘ (P_{1}^{∗} ∘ Q_{1}^{∗} + P_{2}^{∗} ∘ Q_{2}^{∗}) ∘ K ∘ Q_{2})
= T r_{V_{1}}(P_{1} ∘ g^{−1} ∘ P_{1}^{∗} ∘ Q_{1}^{∗} ∘ K ∘ Q_{1})
+T r_{V_{2}}(P_{2} ∘ g^{−1} ∘ P_{2}^{∗} ∘ Q_{2}^{∗} ∘ K ∘ Q_{2})
= T r_{V_{1}}(g_{1}^{−1} ∘ Q_{1}^{∗} ∘ K ∘ Q_{1}) + T r_{V_{2}}(g_{2}^{−1} ∘ Q_{2}^{∗} ∘ K ∘ Q_{2}).

Lemma 3.102. Let V = U ⊕ U^{∗}, with operators P_{i}, Q_{i}. The direct sum induces a symmetric form on V,

(3.8)  P_{1}^{∗} ∘ P_{2} + P_{2}^{∗} ∘ d_{U} ∘ P_{1}.

If U is finite-dimensional, then this symmetric form is a metric.

Proof.

(P_{1}^{∗} ∘ P_{2} + P_{2}^{∗} ∘ d_{U} ∘ P_{1})^{∗} ∘ d_{V} = P_{2}^{∗} ∘ P_{1}^{∗∗} ∘ d_{V} + P_{1}^{∗} ∘ d_{U}^{∗} ∘ P_{2}^{∗∗} ∘ d_{V}
= P_{2}^{∗} ∘ d_{U} ∘ P_{1} + P_{1}^{∗} ∘ d_{U}^{∗} ∘ d_{U^{∗}} ∘ P_{2}
= P_{1}^{∗} ∘ P_{2} + P_{2}^{∗} ∘ d_{U} ∘ P_{1}.
(P_{1}^{∗} ∘ P_{2} + P_{2}^{∗} ∘ d_{U} ∘ P_{1}) ∘ (Q_{1}∘ d_{U}^{−1}∘ Q_{2}^{∗} + Q_{2} ∘ Q_{1}^{∗}) = P_{2}^{∗} ∘ Q_{2}^{∗} + P_{1}^{∗} ∘ Q_{1}^{∗}
= Id_{V^{∗}}.

(Q_{1}∘ d_{U}^{−1}∘ Q_{2}^{∗} + Q_{2} ∘ Q_{1}^{∗}) ∘ (P_{1}^{∗} ∘ P_{2} + P_{2}^{∗} ∘ d_{U} ∘ P_{1}) = Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}
= Id_{V} .

Example 3.103. Given a metric g_{U} on U, if V = U ⊕ U^{∗}, with operators P_{i}, Q_{i}, then the direct sum of the metric g_{U} and its dual d_{U} ∘ g_{U}^{−1} is a metric on V:

g_{U} ⊕ g_{U^{∗}} = P_{1}^{∗} ∘ g_{U} ∘ P_{1} + P_{2}^{∗} ∘ d_{U} ∘ g_{U}^{−1} ∘ P_{2},

as in Theorem 3.17 and Corollary 3.18.

The map K = Q_{2} ∘ g_{U} ∘ P_{1} + Q_{1} ∘ g_{U}^{−1} ∘ P_{2} is an involution on V, as in Equation (1.17) from Theorem 1.130, and it is an isometry with respect to both the above induced metric g_{U} ⊕g_{U^{∗}}, and the canonical metric g_{V} from (3.8) in Lemma 3.102. In particular, if \frac{1}{2} ∈ 𝕂, then Lemma 3.55 applies, so that the direct sum V = V_{1} ⊕ V_{2}, where

P′_{1} = \frac{1}{2}(Id_{V} + K) = \frac{1}{2}(Id_{V} + Q_{2} ∘ g_{U} ∘ P_{1} + Q_{1} ∘ g_{U}^{−1} ∘ P_{2})
P′_{2} = \frac{1}{2}(Id_{V} − K) = \frac{1}{2}(Id_{V} − Q_{2} ∘ g_{U} ∘ P_{1} − Q_{1} ∘ g_{U}^{−1} ∘ P_{2}),

is orthogonal with respect to both metrics on V. Each of the two metrics on V induces a metric on V_{1} and on V_{2}.

Exercise 3.104. For V = U ⊕ U^{∗} and V = V_{1} ⊕ V_{2} as in the above Example, the two induced metrics on V_{1} are identical, while those on V_{2} are opposite.

Hint. It is more convenient to check the equality of the inverses of the induced metrics on V_{1}, using (3.4) from Corollary 3.18 and the formulas from Theorem 3.58:

P′_{1} ∘ g_{V}^{−1} ∘ (P′_{1})^{∗}
= \frac{1}{2}(Id_{V} + K) ∘ (Q_{1}∘ d_{U}^{−1}∘ Q_{2}^{∗} + Q_{2} ∘ Q_{1}^{∗}) ∘ \frac{1}{2}(Id_{V} + K)^{∗}
= P′_{1} ∘ (g_{U} ⊕ g_{U^{∗}})^{−1} ∘ (P′_{1})^{∗}
= \frac{1}{2}(Id_{V} + K) ∘ (Q_{1} ∘ g_{U}^{−1} ∘ Q_{1}^{∗} + Q_{2} ∘ g_{U} ∘ d_{U}^{−1} ∘ Q_{2}^{∗}) ∘ \frac{1}{2}(Id_{V} + K)^{∗}
= \frac{1}{2}(Q_{1} ∘ g_{U}^{−1} ∘ Q_{1}^{∗} + Q_{2} ∘ g_{U}^{∗} ∘ Q_{2}^{∗} + Q_{2} ∘ Q_{1}^{∗} + Q_{1} ∘ d_{U}^{−1} ∘ Q_{2}^{∗})
= \frac{1}{2}(g_{V}^{−1} + (g_{U} ⊕ g_{U^{∗}})^{−1}).

The calculations for the metrics induced on V_{2} are similar.

Example 3.105. Let V = U ⊕ U^{∗}, with operators P_{i}, Q_{i}. The direct sum induces an antisymmetric form on V,

(3.9)  P_{2}^{∗} ∘ d_{U} ∘ P_{1} − P_{1}^{∗} ∘ P_{2}.

If U is finite-dimensional, then this antisymmetric form is symplectic (Definition 3.82). The construction is similar to the induced symmetric form (3.8) from Lemma 3.102, and canonical up to sign (as in (3.6), (3.7)). The inverse of the symplectic form is Q_{1} ∘ d_{U}^{−1} ∘ Q_{2}^{∗} − Q_{2} ∘ Q_{1}^{∗}.

3.7.5. Isotropic maps and graphs.

Definition 3.106. Given a bilinear form g : V → V^{∗}, a linear map A : U → V is \underline{isotropic} with respect to g means that the pullback of g by A is zero:

A^{∗} ∘ g ∘ A = 0_{Hom(U,U^{∗})}.

Exercise 3.107. Given V = V_{1} ⊕ V_{2} with projection and inclusion operators (P_{1}, P_{2}), (Q_{1}, Q_{2}), and a bilinear form h : V → V^{∗}, the following are equivalent.

(1) Q_{1} and Q_{2} are both isotropic with respect to h.

(2) The involution K = Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2} satisfies h = −K^{∗} ∘ h ∘ K. If, further, \frac{1}{2} ∈ 𝕂 and K ∈ End(V) is any involution satisfying h = −K^{∗} ∘ h ∘ K, then the direct sum produced by K has both of the above equivalent properties.

Hint. The expression Q_{1} ∘ P_{1} − Q_{2} ∘ P_{2} is as in Example 1.116.

Exercise 3.108. Given V = V_{1}⊕V_{2} with inclusion operators Q_{i}, bilinear forms g_{1} : V_{1} → V_{1}^{∗}, g_{2} : V_{2} → V_{2}^{∗}, and a map A : V_{1} → V_{2}, the following are equivalent.

(1) g_{1} = A^{∗} ∘ g_{2} ∘ A.

(2) The map Q_{1} + Q_{2} ∘ A : V_{1} → V is isotropic with respect to the bilinear form g_{1} ⊕ (−g_{2}).

Hint. The first property is that g_{1} is the pullback of g_{2} by A as in Definition 3.7; special cases include A being an isometry (Definition 3.23) or a symplectic isometry (Definition 3.85).

The second property refers to the direct sum of bilinear forms as in (3.3) from Notation 3.9. The expression Q_{1} + Q_{2} ∘ A is from the notion that a “graph” of a linear map can be defined in terms of a direct sum, as in Exercise 1.103.

Exercise 3.109. ([LP]) Let V = U ⊕ U^{∗}. Given maps E : W → U and h : U → W^{∗}, the following are equivalent.

(1) The bilinear form h ∘ E : W → W^{∗} is antisymmetric.

(2) The map Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W} : W → V is isotropic with respect to the

symmetric form (3.8) on V from Lemma 3.102.

Further, if E is a linear monomorphism, then so is Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W}.

Hint. By Definition 3.106, the second property is that the pullback of the symmetric form (3.8) on V = U ⊕ U^{∗} from Lemma 3.102 by the map Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W} : W → V is 0_{Hom(W,W^{∗})}. The transpose T_{W} (h ∘ E) is E^{∗} ∘ h^{∗} ∘ d_{W}.

(Q_{1}∘E + Q_{2}∘ h^{∗}∘ d_{W})^{∗}∘(P_{1}^{∗}∘ P_{2} + P_{2}^{∗}∘ d_{U} ∘ P_{1})∘(Q_{1}∘E + Q_{2}∘ h^{∗}∘ d_{W})

= E^{∗} ∘ h^{∗} ∘ d_{W} + d_{W}^{∗} ∘ h^{∗∗} ∘ d_{U} ∘ E
= E^{∗} ∘ h^{∗} ∘ d_{W} + h ∘ E.

For any maps F, G, if

(Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W}) ∘ F = (Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W}) ∘ G,

then

P_{1} ∘ (Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W}) ∘ F = P_{1} ∘ (Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W}) ∘ G
= E ∘ F = E ∘ G,

so if E is a linear monomorphism (Definition 0.42), then F = G, proving the second claim.

If W = U and E = Id_{U}, then this construction is exactly the graph of h^{∗} ∘ d_{U}, as in Exercise 1.103. A generalization of the construction appears in Section 4.2.

Exercise 3.110. ([LP]) Let V = U ⊕ U^{∗}. Given maps E : W → U and h : U → W^{∗}, the following are equivalent.

(1) The bilinear form h ∘ E : W → W^{∗} is symmetric.

(2) The map Q_{1} ∘ E + Q_{2} ∘ h^{∗} ∘ d_{W} : W → V is isotropic with respect to the antisymmetric form (3.9) from Example 3.105.

3.7.6. The adjoint.

Definition 3.111. Metrics g, h, on U, V induce an \underline{adjoint} map,

(3.10)  Hom(h, g^{−1}) ∘ t_{UV} : Hom(U, V) → Hom(V, U) : A ↦ g^{−1} ∘ A^{∗} ∘ h.

Exercise 3.112. Given metrics g and h on U and V, the map (3.10) is an isometry with respect to the induced b metrics. Also, if A : U → V is an isometry, then its adjoint is an isometry V → U.

Hint. The first assertion follows from the fact that g, h, and t_{UV} are isometries. The second claim follows from the following equation, which uses the symmetry of g and h, and Lemma 1.14:

(3.11) (g^{−1} ∘ A^{∗} ∘ h)^{∗} = h^{∗} ∘ A^{∗∗} ∘ (g^{−1})^{∗} = h ∘ d_{V}^{−1} ∘ A^{∗∗} ∘ d_{U} ∘ g^{−1} = h ∘ A ∘ g^{−1},

and the hypothesis g = A^{∗} ∘ h ∘ A:

(g^{−1} ∘ A^{∗} ∘ h)^{∗} ∘ g ∘ (g^{−1} ∘ A^{∗} ∘ h) = (h ∘ A ∘ g^{−1}) ∘ A^{∗} ∘ h
= h ∘ A ∘ A^{−1} = h.

Lemma 3.113. Given metrics g and h on U and V, the composite of adjoint maps,

(Hom(g, h^{−1}) ∘ t_{V U}) ∘ (Hom(h, g^{−1}) ∘ t_{UV}) : Hom(U, V) → Hom(U, V)

is the identity. In particular, the adjoint map Hom(g, g^{−1}) ∘ t_{UU} : End(U) → End(U) is an involution.

Proof. Using (3.11),

h^{−1} ∘ (g^{−1} ∘ A^{∗} ∘ h)^{∗} ∘ g = h^{−1} ∘ (h ∘ A ∘ g^{−1}) ∘ g = A.

Exercise 3.114. Given a metric g on U, if T r_{U}(Id_{U}) ≠ 0, then the adjoint map Hom(g, g^{−1}) ∘ t_{UU} respects any direct sum End(U) = 𝕂 ⊕ End_{0}(U) as in Example 2.9. The restriction of the adjoint map to End_{0}(U) is an involution and an isometry.

Hint. The direct sum refers to the construction of Example 2.9, and it is easily checked that P_{I} ∘ Hom(g, g^{−1}) ∘ t_{UU} ∘ Q_{i} is zero for i ≠ I. The direct sum is orthogonal as in Theorem 3.52, and Theorem 3.61 applies to the map induced by the adjoint on End_{0}(U).

Theorem 3.115. Given a metric g on U, the following diagram is commutative, where s and s′ are switching involutions.

\xymatrix{U ⊗ U \ar[rr]^{s} \ar[d]^{[g⊗g]} &  & U ⊗ U \ar[d]^{[g⊗g]} \\ U^{∗} ⊗ U^{∗} \ar[d]^{k_{UU^{∗}}} \ar[rr]^{s′} &  & U^{∗} ⊗ U^{∗} \ar[d]^{k_{UU^{∗}}} \\ Hom(U, U^{∗}) \ar[r]^{t_{UU^{∗}}} & Hom(U^{∗∗}, U^{∗}) \ar[r]^{Hom(d_{U} ,Id_{U^{∗}})} & Hom(U, U^{∗}) \\ End(U) \ar[u]_{Hom(Id_{U} ,g)} \ar[r]^{t} & End(U^{∗}) \ar[u]_{Hom(g^{∗},Id_{U^{∗}})} \ar[r]^{Hom(g,g^{−1})} & End(U) \ar[u]_{Hom(Id_{U} ,g)} \\ U^{∗} ⊗ U \ar[u]_{k} \ar[r]^{p} & U^{∗∗} ⊗ U^{∗} \ar[u]_{k′} \ar[r]^{[g^{∗}⊗g^{−1}]} & U^{∗} ⊗ U \ar[u]_{k} \\ U ⊗ U \ar[u]_{[g⊗Id_{U}]} \ar[rr]^{s} &  & U ⊗ U \ar[u]_{[g⊗Id_{U}]}}

All the horizontal compositions of arrows define involutions, and if \frac{1}{2} ∈ 𝕂, then they produce orthogonal direct sums on the spaces in the left column.

Proof. The composite in the third row is T_{U}, and the second square from the top does not involve the metric g — it was considered in Lemma 3.6.

The composite in the fifth row, [g^{∗}⊗g^{−1}]∘p, is the only involution not considered earlier. The commutativity of all the squares is easy to check.

The direct sums are produced by the involutions as in Lemma 1.114. The orthogonality of the direct sum for Hom(U, U^{∗}) was checked in Theorem 3.56, and the orthogonality of the other direct sums similarly follows from Lemma 3.55 since all the horizontal arrows are isometries and involutions, or from Theorem 3.61 since all the vertical arrows are isometries which respect the direct sums, by Lemma 1.120. In particular, the direct sum U^{∗} ⊗ U^{∗} = S^{2}(U^{∗}) ⊕ Λ^{2}(U^{∗}) from Lemma 3.6 is orthogonal.

Definition 3.116. Given a metric g on U, if \frac{1}{2} ∈ 𝕂, then the orthogonal direct sum on End(U), produced by the involution Hom(g, g^{−1}) ∘ t as in Theorem 3.115, defines subspaces of \underline{self-adjoint} (A = g^{−1} ∘ A^{∗} ∘ g) and \underline{skew-adjoint} (A = −g^{−1} ∘ A^{∗} ∘ g) endomorphisms.

Example 3.117. Given a metric g on U, if \frac{1}{2} ∈ 𝕂, then the bilinear form h : U → U^{∗} is a symmetric (or, antisymmetric) form if and only if g^{−1} ∘h ∈ End(U) is self-adjoint (respectively, skew-adjoint). This is the action of the middle left vertical arrow, and its inverse, from Theorem 3.115, respecting the direct sums Hom(U, U^{∗}) → End(U).

Exercise 3.118. Given metrics g, h on U, V, if \frac{1}{2} ∈ 𝕂 then for any map A : U → V,

Hom(g^{−1} ∘ A^{∗} ∘ h, A) : End(U) → End(V)

respects the direct sum from Definition 3.116.

Hint. The following diagram is commutative, using Lemma 1.8 and the symmetric property of g and h, so Lemma 1.120 applies.

\xymatrix{End(U) \ar[d]^{Hom(g^{−1}∘A^{∗}∘h,A)} \ar[r]^{t_{UU}} & End(U^{∗}) \ar[r]^{Hom(g,g^{−1})} & End(U) \ar[d]^{Hom(g^{−1}∘A^{∗}∘h,A)} \\ End(V) \ar[r]^{t_{V V}} & End(V^{∗}) \ar[r]^{Hom(h,h^{−1})} & End(V)}

Exercise 3.119. Given a metric g on V, if \frac{1}{2} ∈ 𝕂, then a skew-adjoint A ∈ End(V) satisfies T r_{V} (A) = 0. If, further, T r_{V} (Id_{V}) ≠ 0, then any A ∈ End(V) can be written as a sum of three terms,

A = \frac{T r_{V} (A)}{T r_{V} (Id_{V})} · Id_{V} + A_{1} + A_{2},

where A_{1} and A_{2} have trace 0, A_{1} is self-adjoint, and A_{2} is skew-adjoint.

Hint. The first claim follows from Lemma 2.5 and Lemma 2.6. The second claim is an analogue of Corollary 3.36. Apply Theorem 1.119 to T r_{V} and the involution Hom(g, g^{−1}) ∘ t on End(V) to get a direct sum decomposition.

Exercise 3.120. Given a metric g on U, any scalar α ∈ 𝕂, and any vector u ∈ U, the endomorphism

α · k_{UU}((g(u)) ⊗ u) ∈ End(U)

is self-adjoint. If, further, α·(g(u))(u) = 1, then α·k_{UU}((g(u))⊗u) is an idempotent.

Hint. From the commutativity of the diagram in Theorem 3.115,

(Hom(g, g^{−1}) ∘ t_{UU})(k_{UU}((g(u)) ⊗ u))
= (Hom(g, g^{−1}) ∘ t_{UU} ∘ k_{UU} ∘ [g ⊗ Id_{U}])(u ⊗ u)
= (k_{UU} ∘ [g ⊗ Id_{U}] ∘ s)(u ⊗ u)
= k_{UU}((g(u)) ⊗ u).

The easily checked idempotent property is related to Exercise 2.15.

Exercise 3.121. Given metrics g, h on U, V, any vector u ∈ U, and any map A : U → V, the two self-adjoint endomorphisms from Exercise 3.120 are related by the map from Exercise 3.118:

Hom(g^{−1} ∘ A^{∗} ∘ h, A)(k_{UU}((g(u)) ⊗ u)) = k_{V V} ((h(A(u))) ⊗ (A(u))).

Hint. The left square is commutative by Lemma 1.36, and the right square is commutative by Lemma 1.58 and Equation (3.11).

\xymatrix{U ⊗ U \ar[d]^{[A⊗B]} \ar[r]^{[g⊗Id_{U}]} & U^{∗} ⊗ U \ar[r]^{k_{UU}} \ar[d]^{[(h∘A∘g^{−1})⊗B]} & End(U) \ar[d]^{Hom(g^{−1}∘A^{∗}∘h,B)} \\ V ⊗ V \ar[r]^{[h⊗Id_{V} ]} & V^{∗} ⊗ V \ar[r]^{k_{V V}} & End(V)}

The equality follows from the case where B = A, and starting with u⊗u ∈ U⊗U.

Exercise 3.122. Given a metric g on U, and an endomorphism A ∈ End(U), any pair of two of the following three statements implies the remaining third statement.

(1) A is an involution.

(2) A is self-adjoint.

(3) A is an isometry.

3.7.7. Some formulas from applied mathematics.

Remark 3.123. The following few statements are related to the Householder reflection R.

Exercise 3.124. Given a metric g on U, and an element u ∈ U, if (g(u))(u) ≠ 0, then the endomorphism

R = Id_{U} − \frac{2}{(g(u))(u)} · k_{UU}((g(u)) ⊗ u)

is self-adjoint, an involution, and an isometry.

Hint. The second term is from Exercise 3.120. Lemma 1.117 and Exercise 3.122 apply.

Proposition 3.125. Given a metric g on V and \frac{1}{2} ∈ 𝕂, if u, v ∈ V satisfy (g(u))(u) = (g(v))(v) ≠ 0, then there exists an isometry H ∈ End(V) such that H(u) = v.

Proof. Such an isometry may not be unique; the following construction is not canonical, it depends on two cases.

Case 1. If (g(u + v))(u + v) ≠ 0, then consider the isometry from Exercise 3.124, applied to the vector u + v:

R = Id_{V} − \frac{2}{(g(u + v))(u + v)} · k_{V V} ((g(u + v)) ⊗ (u + v)) :
u ↦ u − \frac{2}{(g(u + v))(u + v)} · (g(u + v))(u) · (u + v)
= u − \frac{2}{2 · (g(u))(u) + 2 · (g(u))(v)} · ((g(u))(u) + (g(v))(u)) · (u + v)
= −v.

Let H = −R.

Case 2. If (g(u + v))(u + v) = 0, the calculation

(g(u + v))(u + v) + (g(u − v))(u − v) = 4 · (g(u))(u),

and the assumption \frac{1}{2} ∈ 𝕂, imply that (g(u − v))(u − v) ≠ 0, so we can use the isometry from Exercise 3.124, applied to the vector u − v:

H = R = Id_{V} − \frac{2}{(g(u − v))(u − v)} · k_{V V} ((g(u − v)) ⊗ (u − v)) :
u ↦ u − \frac{2}{(g(u − v))(u − v)} · (g(u − v))(u) · (u − v)
= u − \frac{2}{2 · (g(u))(u) − 2 · (g(u))(v)} · ((g(u))(u) − (g(v))(u)) · (u − v)
= v.

Exercise 3.126. Given a metric g on V and v ∈ V, if (g(v))(v) ≠ 0, then there exists a direct sum V = 𝕂 ⊕ ker(g(v)) such that any direct sum equivalent to it has the properties that it is orthogonal and for any A ∈ End(V),

T r_{V} (Q_{1} ∘ P_{1} ∘ A) = \frac{(g(v))(A(v))}{(g(v))(v)}.

If, further, \frac{1}{2} ∈ 𝕂, then the direct sum produced by the involution −R, for R as in Exercise 3.124, is such an equivalent direct sum.

Hint. Since g(v) ≠ 0_{V^{∗}}, Lemmas 1.96 and 1.97 give a direct sum V = 𝕂 ⊕ ker(g(v)), which is canonical up to equivalence, as follows. Let Q′_{2} be the inclusion of the subspace ker(g(v)) in V. For any α, β ∈ 𝕂 with α · β · (g(v))(v) = 1, define

Q_{1}^{β} : 𝕂 → V : γ ↦ β · γ · v,
P_{1}^{α} = α · g(v) : V → 𝕂,
P′_{2} = Id_{V} − Q_{1}^{β} ∘ P_{1}^{α} : V → ker(g(v)).

For the orthogonality of the direct sum, it is straightforward to check, using the symmetric property of g, that (P_{1}^{α})^{∗} ∘(Q_{1}^{β})^{∗} ∘g = g ∘Q_{1}^{β} ∘P_{1}^{α}, or that (Q_{1}^{β})^{∗} ∘g ∘Q′_{2} and (Q′_{2})^{∗} ∘ g ∘ Q_{1}^{β} are both zero. This is also a special case of Exercise 3.91 with h = g and H = Q_{1}^{β}.

It is also easy to check that

(3.12)  Q_{1}^{β} ∘ P_{1}^{α} ∘ A = k_{V V} (β · α · ((g(v)) ∘ A) ⊗ v) ∈ End(V),

so by the definition of trace,

(3.13)  T r_{V} (Q_{1}^{β} ∘ P_{1}^{α} ∘ A) = Ev_{V} (β · α · ((g(v)) ∘ A) ⊗ v)
(3.14)  = β · α · (g(v))(A(v)) = \frac{(g(v))(A(v))}{(g(v))(v)},

where the RHS of (3.14) does not depend on the choice of α, β. Further, if operators P_{i}, Q_{i} define any direct sum equivalent to the above orthogonal direct sum, then that direct sum is also orthogonal by Lemma 3.50, and Q_{1}^{β} ∘ P_{1}^{α} = Q_{1} ∘ P_{1} as in Lemma 1.91, so the LHS of (3.13) is invariant under equivalent direct sums.

Finally, setting A = Id_{V} in (3.12) gives R = Id_{V} − 2 · Q_{1}^{β} ∘ P_{1}^{α}. If \frac{1}{2} ∈ 𝕂, then the direct sum produced by −R as in Lemma 1.114 has Q_{1}∘P_{1} = \frac{1}{2} ·(Id_{V} +(−R)) = Q_{1}^{β} ∘ P_{1}^{α} and the direct sums are equivalent. The A = Id_{V} case of (3.14) also gives the formula from Example 2.8.

The above steps did not use the invertibility of g, although the notion of orthogonal direct sum was defined only with respect to an invertible metric g.

Exercise 3.127. Given a metric g on V and a direct sum of the form V = 𝕂⊕U with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}), let v = Q_{1}(1). If the direct sum is orthogonal with respect to g, then it is equivalent to a direct sum V = 𝕂⊕ker(g(v)) from Exercise 3.126. The involution from Lemma 3.54,

−K = −Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2},

coincides with the involution from Exercise 3.124,

R = Id_{V} − \frac{2}{(g(v))(v)} · k_{V V} ((g(v)) ⊗ v)
= Id_{V} − \frac{2}{((g ∘ Q_{1})(1))(Q_{1}(1))} · k_{V V} (((g ∘ Q_{1})(1)) ⊗ (Q_{1}(1))) ∈ End(V).

Hint. First, Lemma 3.71 applies to the orthogonal direct sum: (g(v))(v) = (g(Q_{1}(1)))(Q_{1}(1)) = ν ≠ 0. Using orthogonality again, g ∘ Q_{1} ∘ P_{1} = P_{1}^{∗} ∘ Q_{1}^{∗} ∘ g, so for any w ∈ V,

g(Q_{1}(P_{1}(w))) = P_{1}^{∗}(Q_{1}^{∗}(g(w)))
= P_{1}(w) · g(Q_{1}(1)) = P_{1}(w) · g(v) = g(w) ∘ Q_{1} ∘ P_{1} : v ↦
P_{1}(w) · (g(v))(v) = (g(w))(Q_{1}(P_{1}(v))) = (g(w))(P_{1}(v) · Q_{1}(1))
= P_{1}(Q_{1}(1)) · (g(w))(v) = (g(w))(v)
⟹ P_{1}(w) = \frac{(g(w))(v)}{(g(v))(v)}.

The equivalence of the direct sums follows, using the symmetric property of g:

(Q_{1} ∘ P_{1})(w) = P_{1}(w) · Q_{1}(1) = \frac{(g(w))(v)}{(g(v))(v)} · v,
(Q_{1}^{β} ∘ P_{1}^{α})(w) = α · β · (g(v))(w) · v = \frac{(g(v))(w)}{(g(v))(v)} · v.

The claimed equality also follows, as in (3.12):

−K = Id_{V} − 2 · Q_{1} ∘ P_{1} = Id_{V} − 2 · Q_{1}^{β} ∘ P_{1}^{α} = R.

Exercise 3.128. Given a metric g on V, if T r_{V} (Id_{V}) ≠ 0, then for an orthogonal direct sum End(V) = 𝕂 ⊕ End_{0}(V) with inclusion operator Q_{1}^{β} as in Example 2.9 and Theorem 3.52, the induced metric on 𝕂 is h^{ν}, where ν = β^{2} ·T r_{V} (Id_{V}) does not depend on g. The involution R on End(V) from Exercise 3.127 does not depend on g or β, and reverses the trace: for any A ∈ End(V), T r_{V} (R(A)) = −T r_{V} (A).

Hint. Using Lemma 3.71,

ν = (b(Q_{1}^{β}(1)))(Q_{1}^{β}(1)) = T r_{V} (β · Id_{V} ∘ g^{−1} ∘ (β · Id_{V})^{∗} ∘ g) = β^{2} · T r_{V} (Id_{V}).

For A ∈ End(V),

R(A) = A − \frac{2}{ν} · (b(Q_{1}^{β}(1)))(A) · Q_{1}^{β}(1)
= A − \frac{2}{ν} · T r_{V} (A ∘ g^{−1} ∘ (β · Id_{V})^{∗} ∘ g) · β · Id_{V}
= A − \frac{2 · T r_{V} (A) · β^{2}}{β^{2} · T r_{V} (Id_{V})} · Id_{V} .

Remark 3.129. The following few exercises are related to the \underline{block vec} operation from [O].

In the following diagram,

\xymatrix{ & Hom(U ⊗V, W^{∗}⊗X^{∗}) \ar[dr]^{Hom(Id_{U⊗V} ,j̃)} \\ Hom(U, W^{∗})⊗Hom(V, X^{∗}) \ar[ur]^{j} &  & Hom(U ⊗V, (W ⊗X)^{∗}) \ar[ddd]_{□} \\  & (U ⊗ V)^{∗}⊗(W ⊗ X)^{∗} \ar[ur]^{k_{U⊗V,(W⊗X)^{∗}}} \\ U^{∗}⊗W^{∗}⊗V^{∗}⊗X^{∗} \ar[uu]^{[k_{UW^{∗}} ⊗k_{V X^{∗}}]} \ar[r]^{s_{1}} \ar[dr]_{s_{3}} & U^{∗} ⊗ V^{∗} ⊗ W^{∗} ⊗ X^{∗} \ar[u]^{[j̃⊗j̃]} \ar[d]_{s_{2}} \\  & V^{∗} ⊗ X^{∗} ⊗ U^{∗} ⊗ W^{∗} \ar[d]_{[j̃⊗j̃]} & Hom(V ⊗X, (U ⊗W)^{∗}) \\  & (V ⊗ X)^{∗} ⊗ (U ⊗ W)^{∗} \ar[ur]_{k_{V ⊗X,(U⊗W)^{∗}}}}

s_{1}, s_{2}, and s_{3} = s_{2}∘s_{1} are switching maps, and the various j̃ maps are as in Notation 2.43. The top block is commutative, it is similar to the diagram from Lemma 2.32.

Notation 3.130. If three of the four spaces U, V, W, X are finite-dimensional, then all of the arrows in the above diagram are invertible. Define the map

□ : Hom(U ⊗ V, (W ⊗ X)^{∗}) → Hom(V ⊗ X, (U ⊗ W)^{∗})

to equal the composite going counter-clockwise around the lower right square in the diagram.

Exercise 3.131. ([O] Theorem 1) If three of the four spaces U, V, W, X are finite-dimensional, then for any A ∈ Hom(U ⊗ V, (W ⊗ X)^{∗}), the following are equivalent.

(1) There exist h_{1} ∈ Hom(U, W^{∗}), h_{2} ∈ Hom(V, X^{∗}) such that

A = j̃ ∘ [h_{1} ⊗ h_{2}].
(2) There exist ϕ_{1} ∈ (V ⊗ X)^{∗} and ϕ_{2} ∈ (U ⊗ W)^{∗} such that
□(A) = k_{V ⊗X,(U⊗W)^{∗}}(ϕ_{1} ⊗ ϕ_{2}).

In the special case W = U, X = V, (1) can be re-written using Notation 3.13: (1′) There exist h_{1} ∈ Hom(U, U^{∗}), h_{2} ∈ Hom(V, V^{∗}) such that A = {h_{1} ⊗ h_{2}}.

Exercise 3.132. ([O] Corollary 1) If V is finite-dimensional, then for any A ∈ Hom(V ⊗ V, (V ⊗ V)^{∗}), the following are equivalent.

(1) There exists h ∈ Hom(V, V^{∗}) such that A = {h ⊗ h}.

(2) There exists ϕ ∈ (V ⊗ V)^{∗} such that □(A) = k_{V ⊗V,(V ⊗V)^{∗}}(ϕ ⊗ ϕ).

Either (1) or (2) implies that the bilinear form □(A) is symmetric.

Remark 3.133. The following two Propositions relating the b metric to a trace on a tensor product space are analogous to a formula involving the “commutation matrix” K (from Remark 1.72), which appears in [HJ] §4.3, and [Magnus] (exercise 3.9: trK(A′ ⊗ B) = trA′B).

Proposition 3.134. Given metrics g and h on U and V, for A, B ∈ Hom(U, V),

T r_{V^{∗}⊗U}([(h ∘ d_{V}^{−1}) ⊗ g^{−1}] ∘ p ∘ [B^{∗} ⊗ A]) = T r_{U}(g^{−1} ∘ B^{∗} ∘ h ∘ A).

Proof. In the following diagram,

\xymatrix{U^{∗}⊗V⊗U^{∗}⊗V \ar[d]^{[[(d_{U} ∘g^{−1})⊗h]⊗Id_{U^{∗}⊗V} ]} & U^{∗}⊗V⊗U^{∗}⊗V \ar[d]^{[p⊗Id_{U^{∗}⊗V} ]} \ar[r]^{a_{1}} \ar[l]_{s} & Hom(U,V)⊗Hom(U,V) \ar[d]_{[t_{UV} ⊗Id_{Hom(U,V )}]} \\ U^{∗∗}⊗V^{∗}⊗U^{∗}⊗V \ar[d]^{[j⊗Id_{U^{∗}⊗V} ]} & V^{∗∗}⊗U^{∗}⊗U^{∗}⊗V \ar[r]^{a_{2}} \ar[d]^{[[Id_{V ^{∗∗}} ⊗g^{−1}]⊗[Id_{U^{∗}} ⊗h]]} & Hom(V^{∗},U^{∗})⊗Hom(U,V) \ar[dd]_{j} \\ Hom(U^{∗}⊗V,𝕂⊗𝕂)⊗U^{∗}⊗V \ar[d]^{[Hom(Id_{U^{∗}⊗V},l)⊗Id_{U^{∗}⊗V} ]} & V^{∗∗}⊗U⊗U^{∗}⊗V^{∗} \ar[d]^{s} \\ (U^{∗} ⊗ V)^{∗} ⊗ U^{∗} ⊗ V \ar[dd]_{Ev_{U^{∗}⊗V}} & V^{∗∗}⊗V^{∗}⊗U^{∗}⊗U \ar[ddl]_{l∘[Ev_{V ^{∗}} ⊗Ev_{U}]} \ar[d]^{[k_{V ^{∗}V ^{∗}} ⊗k_{UU}]} & Hom(V^{∗} ⊗ U, U^{∗} ⊗ V) \ar[dd]_{Hom(Id_{V ^{∗}⊗U} ,p)} \\  & End(V^{∗})⊗End(U) \ar[dl]^{l∘[T r_{V ^{∗}}⊗T r_{U}]} \ar[d]^{j_{2}} \\ 𝕂 & End(V^{∗} ⊗ U) \ar[l]^{T r_{V ^{∗}⊗U}} & Hom(V^{∗}⊗U, V^{∗∗}⊗U^{∗}) \ar[l]^{a_{3}}}

the arrow s in the top row switches the two V factors, and the abbreviated arrow labels are

a_{1} = [k_{UV} ⊗ k_{UV}]
a_{2} = [k_{U^{∗}V^{∗}} ⊗ k_{UV}]
a_{3} = Hom(Id_{V^{∗}⊗U}, [(h ∘ d_{V}^{−1}) ⊗ g^{−1}]).

The top right square is commutative by Lemmas 1.36 and 1.71. The lower left triangle is commutative by Corollary 2.36, and the triangle above that by the definition of trace. Starting with Φ ⊗ ϕ ⊗ ξ ⊗ v ∈ V^{∗∗} ⊗ U^{∗} ⊗ U^{∗} ⊗ V, the lower right square is commutative:

Φ⊗ϕ⊗ξ⊗v ↦ (a_{3} ∘ Hom(Id_{V^{∗}⊗U}, p) ∘ j ∘ a_{2})(Φ⊗ϕ⊗ξ⊗v)
= [(h ∘ d_{V}^{−1})⊗g^{−1}] ∘ p ∘ [(k_{U^{∗}V^{∗}}(Φ⊗ϕ))⊗(k_{UV} (ξ⊗v))] :
ψ⊗u ↦ (h((ξ(u)) · v))⊗(g^{−1}((Φ(ψ)) · ϕ)),
Φ⊗ϕ⊗ξ⊗v ↦ (j_{2} ∘ [k_{V^{∗}V^{∗}} ⊗k_{UU}] ∘ s ∘ [[Id_{V^{∗∗}} ⊗g^{−1}]⊗[Id_{U^{∗}} ⊗h]])(Φ⊗ϕ⊗ξ⊗v)
= [(k_{V^{∗}V^{∗}}(Φ⊗(h(v))))⊗(k_{UU}(ξ⊗(g^{−1}(ϕ))))] :
ψ⊗u ↦ (Φ(ψ) · h(v))⊗(ξ(u) · g^{−1}(ϕ)).

Starting with ϕ⊗w⊗ξ⊗v ∈ U^{∗} ⊗V ⊗U^{∗} ⊗V, the upper left square is commutative:

ϕ⊗w⊗ξ⊗v↦(l∘[Ev_{V^{∗}}⊗Ev_{U}]∘s∘[[Id_{V^{∗∗}}⊗g^{−1}]⊗[Id_{U^{∗}}⊗h]]∘[p⊗Id_{U^{∗}⊗V}])(ϕ⊗w⊗ξ⊗v)
= (l ∘ [Ev_{V^{∗}} ⊗ Ev_{U}] ∘ s)((d_{V} (w)) ⊗ (g^{−1}(ϕ)) ⊗ ξ ⊗ (h(v)))
= ((h(v))(w)) · (ξ(g^{−1}(ϕ))),
ϕ⊗w⊗ξ⊗v↦(Ev_{U^{∗}⊗V}∘[(Hom(Id_{U^{∗}⊗V},l)∘j∘[(d_{U} ∘g^{−1})⊗h])⊗Id_{U^{∗}⊗V}]∘s)(ϕ⊗w⊗ξ⊗v)
= Ev_{U^{∗}⊗V} ((l ∘ [((d_{U} ∘ g^{−1})(ϕ)) ⊗ (h(v))]) ⊗ ξ ⊗ w)
= (ξ(g^{−1}(ϕ))) · ((h(v))(w)).

This last quantity is also the result of the tensor product metric:

({(d_{U} ∘ g^{−1}) ⊗ h}(ϕ ⊗ w))(ξ ⊗ v) = (ξ(g^{−1}(ϕ))) · ((h(v))(w)),

from Corollary 3.19. So the claimed equality follows from the commutativity of the diagram, and the fact that k_{UV}^{−1} is an isometry (Theorem 3.41). Starting with B ⊗ A ∈ Hom(U, V) ⊗ Hom(U, V):

LHS = (T r_{V^{∗}⊗U} ∘ a_{3} ∘ Hom(Id_{V^{∗}⊗U}, p) ∘ j ∘ [t_{UV} ⊗ Id_{Hom(U,V)}])(B ⊗ A)
= (Ev_{U^{∗}⊗V}∘[(Hom(Id_{U^{∗}⊗V},l)∘j∘[(d_{U} ∘g^{−1})⊗h])⊗Id_{U^{∗}⊗V}]∘s ∘ a_{1}^{−1})(B ⊗ A)
= ({(d_{U} ∘ g^{−1}) ⊗ h}(k_{UV}^{−1} (B)))(k_{UV}^{−1} (A))
= (b(B))(A) = RHS.

Proposition 3.135. Given metrics g and h on U and V, for A, B ∈ Hom(U, V),

T r_{{(d_{V} ∘h^{−1})⊗g}}(f_{UV} ∘ [B^{∗} ⊗ A]) = T r_{g}(B^{∗} ∘ h ∘ A).

Proof. By Lemma 1.70 and the previous Proposition,

LHS = T r_{V^{∗}⊗U}([(d_{V} ∘ h^{−1}) ⊗ g]^{−1} ∘ j^{−1} ∘ Hom(Id_{V^{∗}⊗U}, l)^{−1} ∘ f_{UV} ∘ [B^{∗}⊗ A])
= T r_{V^{∗}⊗U}([(h ∘ d_{V}^{−1}) ⊗ g^{−1}] ∘ p ∘ [B^{∗} ⊗ A])
= T r_{U}(g^{−1} ∘ B^{∗} ∘ h ∘ A) = RHS.

142 3. BILINEAR FORMS

3.7.8. Eigenvalues.

Exercise 3.136. Suppose h and g are bilinear forms on V, and g is symmetric. If h(v_{1}) = λ_{1} · g(v_{1}), and (T_{V} (h))(v_{2}) = λ_{2} · g(v_{2}), then either λ_{1} = λ_{2}, or (g(v_{1}))(v_{2}) = 0.

Hint.

(λ_{1} − λ_{2}) · (g(v_{1}))(v_{2}) = (λ_{1} · g(v_{1}))(v_{2}) − (λ_{2} · g(v_{2}))(v_{1})
= (h(v_{1}))(v_{2}) − ((T_{V} (h))(v_{2}))(v_{1}) = 0.

Exercise 3.137. Suppose h and g are bilinear forms on V, and g is antisymmetric. If h(v_{1}) = λ_{1} · g(v_{1}), and (T_{V} (h))(v_{2}) = λ_{2} · g(v_{2}), then either λ_{1} = −λ_{2}, or (g(v_{1}))(v_{2}) = 0.

Exercise 3.138. If h and g are both symmetric forms (or both antisymmetric), and h(v_{1}) = λ_{1}·g(v_{1}), and h(v_{2}) = λ_{2}·g(v_{2}), then either λ_{1} = λ_{2}, or (g(v_{1}))(v_{2}) = 0.

Exercise 3.139. If g is a bilinear form on V, and E is an endomorphism of V such that g ∘ E = E^{∗} ∘ g : V → V^{∗}, and E(v_{1}) = λ_{1} · v_{1}, and E(v_{2}) = λ_{2} · v_{2}, then either λ_{1} = λ_{2}, or (g(v_{1}))(v_{2}) = 0.

Hint. When g is a metric, the hypothesis is that E is self-adjoint.

(λ_{1} − λ_{2}) · (g(v_{1}))(v_{2}) = (λ_{1} · g(v_{1}))(v_{2}) − (λ_{2} · g(v_{1}))(v_{2})
= (g(E(v_{1})))(v_{2}) − (g(v_{1}))(E(v_{2}))
= ((g ∘ E)(v_{1}))(v_{2}) − ((E^{∗} ∘ g)(v_{1}))(v_{2}) = 0.

Exercise 3.140. If g is a bilinear form on V, and E is an endomorphism of V such that g ∘ E = −E^{∗} ∘ g : V → V^{∗}, and E(v_{1}) = λ_{1} · v_{1}, and E(v_{2}) = λ_{2} · v_{2}, then either λ_{1} = −λ_{2}, or (g(v_{1}))(v_{2}) = 0. In particular, if \frac{1}{2} ∈ 𝕂, then either λ_{1} = 0, or (g(v_{1}))(v_{1}) = 0.

Hint. This is a skew-adjoint version of the previous Exercise.

Exercise 3.141. Given a metric g on U, a self-adjoint endomorphism H : U → U, and a nonzero element v ∈ U, there exists λ ∈ 𝕂 such that H(v) = λ · v if and only if H commutes with the endomorphism k((g(v)) ⊗ v) from Exercise 3.120.

Hint. The diagram from Exercise 3.121 gives these two equalities:

H ∘ (k((g(v)) ⊗ v)) = k((g(v)) ⊗ (H(v))),
(k((g(v)) ⊗ v)) ∘ H = k((g(H(v))) ⊗ v).

If H(v) = λ · v, then the two quantities are equal. Conversely, if they are equal, then for any u ∈ U,

(k((g(v)) ⊗ (H(v))))(u) = (k((g(H(v))) ⊗ v))(u)
(g(v))(u) · (H(v)) = (g(H(v)))(u) · v.

Since v ≠ 0_{U}, the non-degeneracy of g implies there is some u so that (g(v))(u) ≠ 0. Let λ = \frac{(g(H(v)))(u)}{(g(v))(u)}.

Exercise 3.142. If g is a bilinear form on V, and E is an endomorphism of V such that E^{∗} ∘ g ∘ E = g : V → V^{∗}, and E(v_{1}) = λ_{1} · v_{1}, and E(v_{2}) = λ_{2} · v_{2}, then either λ_{1}·λ_{2} = 1, or (g(v_{1}))(v_{2}) = 0. In particular, either λ_{1}^{2} = 1, or (g(v_{1}))(v_{1}) = 0.

Hint.

(g(v_{1}))(v_{2}) = ((E^{∗} ∘ g ∘ E)(v_{1}))(v_{2}) = (g(E(v_{1})))(E(v_{2}))
= λ_{1} · λ_{2} · (g(v_{1}))(v_{2}).

3.7.9. Canonical metrics.

Example 3.143. Given V finite-dimensional, the canonical invertible map

(k^{∗})^{−1} ∘ e : End(V) → End(V)^{∗}

from Lemma 2.1 is a metric on End(V). It is symmetric by Lemma 1.14 and Lemma 2.1:

((k^{∗})^{−1} ∘ e)^{∗} ∘ d_{End(V)} = e^{∗} ∘ (k^{−1})^{∗∗} ∘ d_{End(V)} = e^{∗} ∘ d ∘ k^{−1} = (k^{∗})^{−1} ∘ e.

This metric on End(V) should be called the \underline{canonical metric}, to distinguish it from the b metric, induced by a choice of metric on V. The non-degeneracy of the metric was considered in Proposition 2.16, where it was also shown that for A, B ∈ End(V),

(3.15)  (((k^{∗})^{−1} ∘ e)(A))(B) = T r_{V} (A ∘ B).

Example 3.144. Given V finite-dimensional, the canonical map f : V^{∗} ⊗ V → (V^{∗}⊗V)^{∗} is invertible, and is symmetric by Lemma 1.67, so it is a metric on V^{∗}⊗V. The dual metric on (V^{∗} ⊗ V)^{∗} is d ∘ f^{−1} = (f^{∗})^{−1}.

This metric on V^{∗} ⊗ V is also canonical, and, in general, different from the tensor product metric induced by a choice of metric on V. By Lemma 1.70, the metric f is equal to the composite Hom(Id_{V^{∗}⊗V} , l) ∘ j ∘ p : V^{∗} ⊗ V → (V^{∗} ⊗ V)^{∗}.

Exercise 3.145. The dual metric d_{𝕂}∘(h^{1})^{−1} on 𝕂^{∗} coincides with the (k^{∗})^{−1}∘e metric on End(𝕂).

Hint. The metric h^{1} is as in Lemma 3.67. For ϕ, ξ ∈ 𝕂^{∗},

((d_{𝕂} ∘ (h^{1})^{−1})(ϕ))(ξ) = ξ(T r_{𝕂}(ϕ)) = ξ(ϕ(1)) = ϕ(1) · ξ(1)
(((k^{∗})^{−1} ∘ e)(ϕ))(ξ) = T r_{𝕂}(ϕ ∘ ξ) = ϕ(ξ(1)) = ξ(1) · ϕ(1).

Exercise 3.146. With respect to the canonical metrics (k^{∗})^{−1} ∘ e on End(𝕂) and h^{1} on 𝕂, T r_{𝕂} is an isometry.

Hint. The canonical metric, applied to A, B ∈ End(𝕂), is:

(((k^{∗})^{−1} ∘ e)(A))(B) = T r_{𝕂}(A ∘ B) = (A ∘ B)(1) = A(B(1)) = B(1) · A(1).

This coincides with the pullback:

(h^{1}(T r_{𝕂}(A)))(T r_{𝕂}(B)) = (h^{1}(A(1)))(B(1)) = A(1) · B(1).

Exercise 3.147. Given finite-dimensional V, the canonical map k : V^{∗} ⊗ V → End(V) is an isometry with respect to the canonical metrics f and (k^{∗})^{−1} ∘ e.

Hint. The pullback of (k^{∗})^{−1} ∘ e by k agrees with f:

k^{∗} ∘ (k^{∗})^{−1} ∘ e ∘ k = f.

Exercise 3.148. Given finite-dimensional V, the canonical map e : End(V) → (V^{∗}⊗V)^{∗} is an isometry with respect to the canonical metrics (k^{∗})^{−1}∘e and (f^{∗})^{−1}.

Hint. The pullback of (f^{∗})^{−1} by e is:

e^{∗} ∘ (f^{∗})^{−1} ∘ e = e^{∗} ∘ (e^{∗})^{−1} ∘ (k^{∗})^{−1} ∘ e = (k^{∗})^{−1} ∘ e.

It follows that f_{V V} is an isometry, but this also follows from Theorem 3.26.

Exercise 3.149. ([G_{2}] §I.8) Given finite-dimensional U, V, if A : U → V is invertible, then Hom(A^{−1}, A) : End(U) → End(V) is an isometry with respect to the (k^{∗})^{−1} ∘ e metrics.

Hint. From the Proof of Lemma 2.6:

(k^{′∗})^{−1} ∘ e′ = (k^{′∗})^{−1} ∘ e′ ∘ Hom(A, A^{−1}) ∘ Hom(A^{−1}, A)
= Hom(A^{−1}, A)^{∗} ∘ (k^{∗})^{−1} ∘ e ∘ Hom(A^{−1}, A).

Exercise 3.150. For U, V, and invertible A as in the previous Exercise, [(A^{−1})^{∗} ⊗ A] : U^{∗} ⊗ U → V^{∗} ⊗ V is an isometry with respect to f_{UU} and f_{V V}.

Hint. This follows from Lemma 1.58:

[(A^{−1})^{∗} ⊗ A] = k_{V V}^{−1} ∘ Hom(A^{−1}, A) ∘ k_{UU},

and could also be checked directly.

Exercise 3.151. Given finite-dimensional V, the transpose t : End(V) → End(V^{∗}) : A ↦ A^{∗} is an isometry with respect to the canonical (k^{∗})^{−1} ∘ e metrics.

Hint. From the Proof of Lemma 2.5:

t^{∗} ∘ (k^{′∗})^{−1} ∘ e′ ∘ t = (k^{∗})^{−1} ∘ e.

Exercise 3.152. Given finite-dimensional U, V, the map j : End(U)⊗End(V) → End(U ⊗ V) is an isometry with respect to the tensor product of canonical metrics, and the canonical metric on End(U ⊗ V).

Hint. By Corollary 2.36,

T r_{U⊗V} ((j(A_{1} ⊗ B_{1})) ∘ (j(A_{2} ⊗ B_{2}))) = T r_{U⊗V} (j((A_{1} ∘ A_{2}) ⊗ (B_{1} ∘ B_{2})))
= T r_{U}(A_{1} ∘ A_{2}) · T r_{V} (B_{1} ∘ B_{2}).

Exercise 3.153. Given finite-dimensional U, if T r_{U}(Id_{U}) ≠ 0, then a direct sum End(U) = 𝕂 ⊕ End_{0}(U) from Example 2.9 is orthogonal with respect to the canonical metric (k^{∗})^{−1} ∘ e on End(U), and this induces a canonical metric on End_{0}(U). The involution from Exercise 3.124, defined in terms of the canonical metric and the canonical element Id_{U}, is given for A ∈ End(U) by:

R : A ↦ A − 2 · \frac{T r_{U}(A)}{T r_{U}(Id_{U})} · Id_{V} ,

which is the same as the involution −K from Lemma 3.54 and the involution R from Exercise 3.128.

Hint. The orthogonality is easy to check; this is also a special case of Exercises 3.126 and 3.127.

Exercise 3.154. Given a metric g on U, the adjoint involution Hom(g, g^{−1}) ∘ t_{UU} on End(U) is an isometry with respect to the canonical metric. If \frac{1}{2} ∈ 𝕂, then the direct sum decomposition into self-adjoint and skew-adjoint endomorphisms, from Definition 3.116, is orthogonal with respect to the canonical metric. On the space of self-adjoint endomorphisms, the metric induced by the canonical metric coincides with the metric induced by the induced b metric. On the space of skewadjoint endomorphisms, the two induced metrics are opposite.

Exercise 3.155. For any bilinear form g : End(V) → End(V)^{∗} (and in particular, any metric g on End(V)), there exists F ∈ End(End(V)) so that for all A, B ∈ End(V),

(g(A))(B) = T r_{V} ((F (A)) ∘ B).
Hint. Define F = e^{−1} ∘ k^{∗} ∘ g. Then by Proposition 2.17,
(g(A))(B) = T r_{V} ((e^{−1}(k^{∗}(g(A)))) ∘ B).

The canonical metric on End(V) from Example 3.143 is the case F = Id_{End(V)}. The b metric from Definition 3.40 induced by a metric h on V,

(b(A))(B) = T r_{V} (h^{−1} ∘ A^{∗} ∘ h ∘ B),

is the case where F is the adjoint involution from Definition 3.111 and Lemma 3.113.

Example 3.156. For the generalized transpose from Definition 1.7 and Example 1.49,

t_{UV}^{W} ∈ Hom(Hom(U, V), Hom(Hom(V, W), Hom(U, W))),

and any bilinear form g : Hom(U, W) → Hom(U, W)^{∗}, the map

Hom(Id_{Hom(U,V)}, Hom(Id_{Hom(V,W)}, g))

transforms t_{UV}^{W} to the scalar valued trilinear form

Hom(Id_{Hom(V,W)}, g) ∘ t_{UV}^{W} ∈ Hom(Hom(U, V), Hom(Hom(V, W), (Hom(U, W))^{∗})).

For A ∈ Hom(U, V), B ∈ Hom(V, W), and C ∈ Hom(U, W),

Hom(Id_{Hom(V,W)}, g) ∘ t_{UV}^{W} : A ↦ g ∘ (t_{UV}^{W} (A)) = g ∘ Hom(A, Id_{U}) :
B ↦ g(B ∘ A) :
C ↦ (g(B ∘ A))(C).

In the special case where g is the metric b from Definition 3.40 induced by metrics g_{1} on U and g_{2} on W,

(g(B ∘ A))(C) = T r_{U}(g_{1}^{−1} ∘ (B ∘ A)^{∗} ∘ g_{2} ∘ C)
= T r_{U^{∗}}(A^{∗} ∘ B^{∗} ∘ g_{2} ∘ C ∘ g_{1}^{−1}).

In a different special case where W = U and g is the canonical metric (k^{∗})^{−1} ∘ e on End(U) from Example 3.143,

(g(B ∘ A))(C) = T r_{U}(B ∘ A ∘ C).

Example 3.157. For any metrics g and h on End(V), consider the b metric from Definition 3.40 induced by g and h on End(End(V)),

(b(E))(F) = T r_{End(V)}(F ∘ g^{−1} ∘ E^{∗} ∘ h),

for E, F ∈ End(End(V)). The canonical metric on End(End(V)),

(3.16)  (k_{End(V),End(V)}^{∗})^{−1} ∘ e_{End(V),End(V)},

is not necessarily the same as the b metric. The metrics can be shown to be different by example, if there exist A, B ∈ End(V), Ψ, Φ ∈ End(V)^{∗} such that (h(A))(B) = 0 and Ψ(A) ≠ 0 and Φ(B) ≠ 0. From Equation (3.5) and Equation (3.15),

(b(k_{End(V),End(V)}(Φ ⊗ A)))(k_{End(V),End(V)}(Ψ ⊗ B))
= Ψ(g^{−1}(Φ))) · (h(A))(B) = 0.

The canonical metric applied to the same inputs has output

T r_{End(V)}((k_{End(V),End(V)}(Φ ⊗ A)) ∘ (k_{End(V),End(V)}(Ψ ⊗ B)))
= T r_{End(V)}(Φ(B) · k_{End(V),End(V)}(Ψ ⊗ A))
= Φ(B) · Ψ(A) ≠ 0.

So even for g and h as in Example 3.143, the canonical metric is not the same as the metric canonically induced by canonical metrics!

CHAPTER 4 Vector Valued Bilinear Forms

The notion of a bilinear form h : V → V^{∗} can be generalized from the “scalar valued” case to a “vector valued” (or “W -valued,” or “twisted”) form h : V → Hom(V, W), so that for inputs v_{1}, v_{2} ∈ V, the output (h(v_{1}))(v_{2}) is an element of W. In the same way as Example 1.51, vector valued bilinear functions B : V × V ⇝ W correspond to W -valued bilinear forms on V, elements of the space Hom(V, Hom(V, W)). Most of the properties of the scalar valued case generalize, but some of the canonical maps are different.

4.1. Transpose for vector valued forms

There would appear to be multiple ways to use the already considered canonical maps to define a transpose operation that switches the inputs for a W -valued form. One way would be to transform Hom(V, Hom(V, W)) into Hom(V, V^{∗}) ⊗ W, and then apply [T_{V} ⊗ Id_{W}], where T_{V} is the transpose for scalar valued forms from Definition 3.2. Another way would be to start from scratch with canonical maps from Chapter 1, which is the approach taken with Lemma 4.1 and Definition 4.2. Of course, these two ways end up with the same result, as shown in Lemma 4.5.

The following Lemma considers a more general domain Hom(V_{1}, Hom(V_{2}, W)), where V_{1} and V_{2} are not necessarily the same. The d map in the diagram is a generalized double duality from Definition 1.13, the t map is a generalized transpose from Definition 1.7, the canonical q maps are as in Definition 1.44, and s is a switching map.

Lemma 4.1. For any V_{1}, V_{2}, W, the following diagram is commutative.

\xymatrix{Hom(V_{1}, Hom(V_{2}, W)) \ar[r]^{q_{1}} \ar[d]^{t_{V_{1},Hom(V_{2},W)}^{W}} & Hom(V_{1} ⊗ V_{2}, W) \ar[dd]^{Hom(s,Id_{W})} \\ Hom(Hom(Hom(V_{2}, W), W), Hom(V_{1}, W)) \ar[d]^{Hom(d_{V_{2}W} ,Id_{Hom(V_{1},W)})} \\ Hom(V_{2}, Hom(V_{1}, W)) \ar[r]^{q_{2}} & Hom(V_{2} ⊗ V_{1}, W)}

Proof. For u ∈ V_{1}, v ∈ V_{2}, and A ∈ Hom(V_{1}, Hom(V_{2}, W)),

(Hom(s, Id_{W}) ∘ q_{1})(A) :
v ⊗ u ↦ ((q_{1}(A)) ∘ s)(v ⊗ u) = (q_{1}(A))(u ⊗ v) = (A(u))(v),
(q_{2} ∘ Hom(d_{V_{2}W} , Id_{Hom(V_{1},W)}) ∘ t_{V_{1},Hom(V_{2},W)}^{W})(A) :
v ⊗ u ↦ (q_{2}((t_{V_{1},Hom(V_{2},W)}^{W}(A)) ∘ d_{V_{2}W}))(v ⊗ u)
= ((t_{V_{1},Hom(V_{2},W)}^{W}(A))(d_{V_{2}W} (v)))(u) = ((d_{V_{2}W} (v)) ∘ A)(u) = (A(u))(v).

Definition 4.2. Corresponding to the left column in the above diagram, let

(4.1)  T_{V_{1},V_{2};W} = Hom(d_{V_{2}W} , Id_{Hom(V_{1},W)}) ∘ t_{V_{1},Hom(V_{2},W)}^{W}.

Notation 4.3. In the special case V = V_{1} = V_{2}, abbreviate T_{V,V;W} = T_{V;W}. In the case W = 𝕂, T_{V;𝕂} is exactly T_{V} from Definition 3.2.

The above expression (4.1) for T_{V_{1},V_{2};W} (and T_{V;W}) uses only Hom spaces and maps from Section 1.1, without referring to tensor products, the scalar field 𝕂, scalar multiplication, or any dual space like V^{∗}. The spaces and s and q maps in Lemma 4.1 use tensor products but no scalars.

Lemma 4.4. For any vector spaces V_{1}, V_{2}, W, T_{V_{1},V_{2};W} is invertible. In particular, for V_{1} = V_{2}, T_{V_{1};W} is an involution on Hom(V_{1}, Hom(V_{1}, W)).

Proof. The first claim follows from Lemma 4.1 and the invertibility of the q maps (Lemma 1.45), and the diagram also shows that

(4.2)  T_{V_{2},V_{1};W} = T_{V_{1},V_{2};W}^{−1} .

The second claim is a special case of (4.2). For h : V_{1} → Hom(V_{2}, W), and v ∈ V_{1}, u ∈ V_{2}, it follows from Definition 4.2 that

(4.3)  ((T_{V_{1},V_{2};W} (h))(u))(v) = (h(v))(u),

as in (3.1) from Lemma 3.3. Instead of using Lemma 4.1 or (4.3), Equation (4.2) can be checked directly from Definition 4.2, using Lemma 1.6, Lemma 1.14, and Lemma 1.17:

T_{V_{2},V_{1};W} (T_{V_{1},V_{2};W} (h)) = Hom((t_{V_{1},Hom(V_{2},W)}^{W}(h)) ∘ d_{V_{2}W} , Id_{W}) ∘ d_{V_{1}W}
= Hom(d_{V_{2}W} , Id_{W}) ∘ Hom(Hom(h, Id_{W}), Id_{W}) ∘ d_{V_{1}W}
= Hom(d_{V_{2}W} , Id_{W}) ∘ d_{Hom(V_{2},W),W} ∘ h
= h.

Relabeling the subscripts then gives the composite in the other order.

The following Lemma uses a canonical n map from Definition 1.39, so that for g ⊗ w ∈ Hom(V, V^{∗}) ⊗ W, (n(g ⊗ w))(v) = (g(v)) ⊗ w.

Lemma 4.5. For any V, W, the following diagram is commutative. If V or W is finite-dimensional, then the k_{V W} and n maps in the diagram are invertible.

\xymatrix{Hom(V,Hom(V,W)) \ar[dd]^{T_{V ;W}} \ar[dr]^{t_{V,Hom(V,W)}^{W}} &  & Hom(V,V^{∗}⊗W) \ar[ll]^{Hom(Id_{V} ,k_{V W})} & Hom(V,V^{∗})⊗W \ar[dd]_{[T_{V} ⊗Id_{W}]} \ar[l]^{n} \\  & Hom(Hom(Hom(V, W), W), Hom(V, W)) \ar@{-->}[dl]^{Hom(d_{V W} ,Id_{Hom(V,W)})} \\ Hom(V,Hom(V,W)) &  & Hom(V,V^{∗}⊗W) \ar[ll]^{Hom(Id_{V} ,k_{V W})} & Hom(V,V^{∗})⊗W \ar[l]^{n}}

Proof. The left triangle is Definition 4.2 for T_{V;W}. For the right part of the diagram, starting with g⊗w ∈ Hom(V, V^{∗}) ⊗ W,

g ⊗ w ↦ (Hom(d_{V W},Id_{Hom(V,W)})∘t_{V,Hom(V,W)}^{W}∘Hom(Id_{V},k_{V W})∘n)(g⊗w):
v ↦ ((t_{V,Hom(V,W)}^{W}(k_{V W} ∘ (n(g ⊗ w)))) ∘ d_{V W})(v)
= (d_{V W} (v)) ∘ k_{V W} ∘ (n(g ⊗ w)) :
u ↦ (d_{V W} (v))(k_{V W} ((g(u)) ⊗ w))
= (k_{V W} ((g(u)) ⊗ w))(v)=(g(u))(v) · w,
g⊗w ↦ (Hom(Id_{V} , k_{V W}) ∘ n ∘ [T_{V} ⊗ Id_{W}])(g ⊗ w) :
v ↦ k_{V W} ((n((T_{V} (g)) ⊗ w))(v))
= k_{V W} (((T_{V} (g))(v)) ⊗ w) :
u ↦ ((T_{V} (g))(v))(u) · w =(g(u))(v) · w.

The invertibility of the canonical maps was stated in Lemma 1.60 and Lemma 1.43.

Lemma 4.6. For any vector spaces U_{1}, U_{2}, V_{1}, V_{2}, W_{1}, W_{2}, and any maps E : U_{1} → V_{1}, F : U_{2} → V_{2}, G : W_{1} → W_{2}, the following diagram is commutative.

\xymatrix{Hom(V_{1}, Hom(V_{2}, W_{1})) \ar[r]^{T_{V_{1},V_{2};W_{1}}} \ar[d]_{Hom(E,Hom(F,G))} & Hom(V_{2}, Hom(V_{1}, W_{1})) \ar[d]^{Hom(F,Hom(E,G))} \\ Hom(U_{1}, Hom(U_{2}, W_{2})) \ar[r]^{T_{U_{1},U_{2},W_{2}}} & Hom(U_{2}, Hom(U_{1}, W_{2}))}

Proof. The claim could be checked by calculating how the composites act on pairs of input vectors, as in Equation (4.3) from the Proof of Lemma 4.4. The following proof instead shows how the claim follows from only the elementary properties of the t and d maps.

The diagram can be expanded using Definition 4.2 and Lemma 1.6:

\xymatrix{Hom(V_{1}, Hom(V_{2}, W_{1})) \ar[rr]^{T_{V_{1},V_{2};W_{1}}} \ar[d]_{Hom(Id_{V_{1}} ,Hom(Id_{V_{2}} ,G))} &  & Hom(V_{2}, Hom(V_{1}, W_{1})) \ar[d]_{Hom(Id_{V_{2}} ,Hom(Id_{V_{1}} ,G))} \\ Hom(V_{1}, Hom(V_{2}, W_{2})) \ar[r]_{t_{V_{1},Hom(V_{2},W_{2})}^{W_{2}}} \ar[d]_{Hom(E,Hom(F,Id_{W_{2}} ))} & M_{1} \ar[d]_{a_{1}} \ar[r]_{Hom(d_{V_{2}W_{2}} ,Id_{Hom(V_{1},W_{2})})} & Hom(V_{2}, Hom(V_{1}, W_{2})) \ar[d]_{Hom(F,Hom(E,Id_{W_{2}} ))} \\ Hom(U_{1}, Hom(U_{2}, W_{2})) \ar[r]_{t_{U_{1},Hom(U_{2},W_{2})}^{W_{2}}} & M_{2} \ar[r]_{Hom(d_{U_{2}W_{2}} ,Id_{Hom(U_{1},W_{2})})} & Hom(U_{2}, Hom(U_{1}, W_{2}))}

where

M_{1} = Hom(Hom(Hom(V_{2}, W_{2}), W_{2}), Hom(V_{1}, W_{2}))
M_{2} = Hom(Hom(Hom(U_{2}, W_{2}), W_{2}), Hom(U_{1}, W_{2}))
a_{1} = Hom(Hom(Hom(F, Id_{W_{2}}), Id_{W_{2}}), Hom(E, Id_{W_{2}})).

The lower left square is commutative by Lemma 1.8. The lower right square is commutative by Lemma 1.6 and Lemma 1.14. These steps are analogous to the steps in the Proof of Lemma 3.8.

The commutativity of the upper block states that for h : V_{1} → Hom(V_{2}, W_{1}),

T_{V_{1},V_{2};W_{2}}(Hom(Id_{V_{2}}, G) ∘ h) = Hom(Id_{V_{1}}, G) ∘ (T_{V_{1},V_{2};W_{1}}(h)).

The following diagram expands the upper block of the previous diagram, so that the compositions down the left and right sides are T_{V_{1},V_{2};W_{1}} and T_{V_{1},V_{2};W_{2}} from Definition 4.2, and the claim of the Lemma follows from the commutativity of the diagram.

\xymatrix{Hom(V_{1}, Hom(V_{2}, W_{1})) \ar[rrrr]^{Hom(Id_{V_{1}} ,Hom(Id_{V_{2}} ,G))} \ar[ddrr]^{t_{V_{1},Hom(V_{2},W_{1})}^{W_{2}}} \ar[ddddr]^{t_{V_{1},Hom(V_{2},W_{1})}^{W_{1}}} &  &  &  & Hom(V_{1}, Hom(V_{2}, W_{2})) \ar[dl]_{t_{V_{1},Hom(V_{2},W_{2})}^{W_{2}}} \\  &  &  & Hom(Hom(Hom(V_{2}, W_{2}), W_{2}), Hom(V_{1}, W_{2})) \ar[ddddr]_{Hom(d_{V_{2}W_{2}} ,Id_{Hom(V_{1},W_{2})})} \\  &  & Hom(Hom(Hom(V_{2},W_{1}),W_{2}), Hom(V_{1},W_{2})) \ar[ur]^{a_{2}} \ar[d]_{a_{3}} \\  &  & Hom(Hom(Hom(V_{2},W_{1}),W_{1}), Hom(V_{1},W_{2})) \ar[ddrr]_{Hom(d_{V_{2}W_{1}} ,Id_{Hom(V_{1},W_{2})})} \\  & Hom(Hom(Hom(V_{2}, W_{1}), W_{1}), Hom(V_{1}, W_{1})) \ar[ur]^{a_{4}} \ar[dl]^{Hom(d_{V_{2}W_{1}} ,Id_{Hom(V_{1},W_{1})})} \\ Hom(V_{2}, Hom(V_{1}, W_{1})) \ar[rrrr]_{Hom(Id_{V_{2}} ,Hom(Id_{V_{1}} ,G))} &  &  &  & Hom(V_{2}, Hom(V_{1}, W_{2}))}

The inside arrows are:

a_{2} = Hom(Hom(Hom(Id_{V_{2}}, G), Id_{W_{2}}), Id_{Hom(V_{1},W_{2})})
a_{3} = Hom(Hom(Id_{Hom(V_{2},W_{1})}, G), Id_{Hom(V_{1},W_{2})})
a_{4} = Hom(Id_{Hom(Hom(V_{2},W_{1}),W_{1})}, Hom(Id_{V_{1}}, G)).

The upper square and the left square are both commutative by Lemma 1.8. The lower square is commutative by Lemma 1.6. The commutativity of the right block follows from Lemma 1.6 and this special case of Lemma 1.14:

Hom(Id_{Hom(V_{2},W_{1})}, G) ∘ d_{V_{2}W_{1}} = Hom(Hom(Id_{V_{2}}, G), Id_{W_{2}}) ∘ d_{V_{2}W_{2}}.

Similarly to Lemma 4.5, the following few Lemmas use canonical n maps — all labeled n, even when some spaces appear in a different order, as in Notation 1.40, so their domain, target, and formula are as indicated by their position in the diagram.

Lemma 4.7. For any U, V_{1}, V_{2}, W, the following diagram is commutative. If U is finite-dimensional, or V_{1} and V_{2} are both finite-dimensional, then all the maps in the diagram are invertible.

\xymatrix{Hom(V_{1}, U ⊗ Hom(V_{2}, W)) \ar[r]^{Hom(Id_{V_{1}} ,n_{1})} & Hom(V_{1}, Hom(V_{2}, W ⊗ U)) \ar[ddd]^{T_{V_{1},V_{2};W ⊗U}} \\ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U \ar[u]_{n_{2}} \ar[d]^{[T_{V_{1},V_{2},W} ⊗Id_{U}]} \\ Hom(V_{2}, Hom(V_{1}, W)) ⊗ U \ar[d]^{n_{3}} \\ Hom(V_{2}, Hom(V_{1}, W) ⊗ U) \ar[r]^{Hom(Id_{V_{2}} ,n_{4})} & Hom(V_{2}, Hom(V_{1}, W ⊗ U))}

Proof. Replacing the T_{V_{1},V_{2};W} and T_{V_{1},V_{2};W ⊗U} downward arrows in the above diagram by composites involving the invertible q maps from Lemma 4.1 gives the following diagram.

\xymatrix{Hom(V_{1}, U ⊗ Hom(V_{2}, W)) \ar[r]^{Hom(Id_{V_{1}} ,n_{1})} & Hom(V_{1}, Hom(V_{2}, W ⊗ U)) \ar[dd]^{q_{3}} \\ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U \ar[u]_{n_{2}} \ar[d]^{[q_{1}⊗Id_{U}]} \\ Hom(V_{1} ⊗ V_{2}, W) ⊗ U \ar[r]_{n_{5}} \ar[d]^{[Hom(s,Id_{W})⊗Id_{U}]} & Hom(V_{1} ⊗ V_{2}, W ⊗ U) \ar[d]^{Hom(s,Id_{W ⊗U})} \\ Hom(V_{2} ⊗ V_{1}, W) ⊗ U \ar[r]_{n_{6}} & Hom(V_{2} ⊗ V_{1}, W ⊗ U) \\ Hom(V_{2}, Hom(V_{1}, W)) ⊗ U \ar[d]^{n_{3}} \ar[u]_{[q_{2}⊗Id_{U}]} \\ Hom(V_{2}, Hom(V_{1}, W) ⊗ U) \ar[r]^{Hom(Id_{V_{2}} ,n_{4})} & Hom(V_{2}, Hom(V_{1}, W ⊗ U)) \ar[uu]_{q_{4}}}

The middle block is commutative by Lemma 1.41. To check the top square, for h ⊗ u ∈ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U, v ⊗ x ∈ V_{1} ⊗ V_{2},

(q_{3} ∘ Hom(Id_{V_{1}}, n_{1}) ∘ n_{2})(h ⊗ u) : v ⊗ x ↦ (q_{3}(n_{1} ∘ (n_{2}(h ⊗ u))))(v ⊗ x)
= ((n_{1} ∘ (n_{2}(h ⊗ u)))(v))(x)
= (n_{1}(u ⊗ (h(v))))(x)
= ((h(v))(x)) ⊗ u,
(n_{5} ∘ [q_{1} ⊗ Id_{U}])(h ⊗ u) : v ⊗ x ↦ (n_{5}((q_{1}(h)) ⊗ u))(v ⊗ x)
= ((q_{1}(h))(v ⊗ x)) ⊗ u
= ((h(v))(x)) ⊗ u.

The lowest square is analogous, with some re-ordering.

The composite Hom(Id_{V_{1}}, n_{1}) ∘ n_{2} appearing in the upper left corner of the diagram from Lemma 4.7 is equal to a composite of the following form, using different versions of the n maps, as in Notation 1.40:

(4.4)  \xymatrix{Hom(V_{1}, Hom(V_{2}, W) ⊗ U) \ar[r] & Hom(V_{1}, Hom(V_{2}, W ⊗ U)) \\ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U \ar[u]}

The spaces can be re-ordered in various ways to state results analogous to Lemma 4.7, with other versions of the n maps but essentially the same Proof. The following Lemma 4.8 is an analogue of Lemma 4.7 but with a longer composite of n maps.

Lemma 4.8. For any U_{1}, U_{2}, V_{1}, V_{2}, W, the following diagram is commutative.

\xymatrix{U_{1} ⊗ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U_{2} \ar[r]^{[Id_{U_{1}} ⊗[T_{V_{1},V_{2};W} ⊗Id_{U_{2}} ]]} \ar[d]^{[Id_{U_{1}} ⊗n_{2}]} & U_{1} ⊗ Hom(V_{2}, Hom(V_{1}, W)) ⊗ U_{2} \ar[d]_{[Id_{U_{1}} ⊗n_{3}]} \\ U_{1} ⊗ Hom(V_{1}, U_{2} ⊗ Hom(V_{2}, W)) \ar[d]^{n_{7}} & U_{1} ⊗ Hom(V_{2}, Hom(V_{1}, W) ⊗ U_{2}) \ar[d]_{n_{8}} \\ Hom(V_{1}, U_{1} ⊗ U_{2} ⊗ Hom(V_{2}, W)) \ar[d]^{Hom(Id_{V_{1}} ,[Id_{U_{1}} ⊗n_{1}])} & Hom(V_{2}, U_{1} ⊗ Hom(V_{1}, W) ⊗ U_{2}) \ar[d]_{Hom(Id_{V_{2}} ,[Id_{U_{1}} ⊗n_{4}])} \\ Hom(V_{1}, U_{1} ⊗ Hom(V_{2}, W ⊗ U_{2})) \ar[d]^{Hom(Id_{V_{1}} ,n_{9})} & Hom(V_{2}, U_{1} ⊗ Hom(V_{1}, W ⊗ U_{2})) \ar[d]_{Hom(Id_{V_{2}} ,n_{10})} \\ Hom(V_{1}, Hom(V_{2}, U_{1} ⊗ W ⊗ U_{2})) \ar[r]_{T_{V_{1},V_{2};U_{1}⊗W ⊗U_{2}}} & Hom(V_{2}, Hom(V_{1}, U_{1} ⊗ W ⊗ U_{2}))}

Proof. As remarked after the Proof of Lemma 4.7, the vertical composites of n maps could be re-arranged into composites of different n maps with the spaces in different order as in (4.4). This could be used to prove this Lemma as stated or to state analogous results.

The following Proof is more direct, and analogous to that of Lemma 4.7, replacing the T maps with composites involving the q maps from Lemma 4.1. In the statement of the Lemma and the following diagram, the maps labeled n_{1}, …, n_{4}, q_{1}, q_{2} are the same that appear in Lemma 4.7.

\xymatrix{U_{1} ⊗ Hom(V_{1} ⊗ V_{2}, W) ⊗ U_{2} \ar[r]^{[Id_{U_{1}} ⊗[Hom(s,Id_{W})⊗Id_{U_{2}} ]]} & U_{1} ⊗ Hom(V_{2} ⊗ V_{1}, W) ⊗ U_{2} \\ U_{1} ⊗ Hom(V_{1}, Hom(V_{2}, W)) ⊗ U_{2} \ar[r]^{[Id_{U_{1}} ⊗[T_{V_{1},V_{2};W} ⊗Id_{U_{2}} ]]} \ar[u]_{[Id_{U_{1}} ⊗[q_{1}⊗Id_{U_{2}} ]]} \ar[d]^{n_{7}∘[Id_{U_{1}} ⊗n_{2}]} & U_{1} ⊗ Hom(V_{2}, Hom(V_{1}, W)) ⊗ U_{2} \ar[u]^{[Id_{U_{1}} ⊗[q_{2}⊗Id_{U_{2}} ]]} \ar[d]_{n_{8}∘[Id_{U_{1}} ⊗n_{3}]} \\ Hom(V_{1}, U_{1} ⊗ U_{2} ⊗ Hom(V_{2}, W)) \ar[d]^{Hom(Id_{V_{1}} ,n_{9}∘[Id_{U_{1}} ⊗n_{1}])} & Hom(V_{2}, U_{1} ⊗ Hom(V_{1}, W) ⊗ U_{2}) \ar[d]_{Hom(Id_{V_{2}} ,n_{10}∘[Id_{U_{1}} ⊗n_{4}])} \\ Hom(V_{1}, Hom(V_{2}, U_{1} ⊗ W ⊗ U_{2})) \ar[d]^{q_{5}} \ar[r]_{T_{V_{1},V_{2};U_{1}⊗W ⊗U_{2}}} & Hom(V_{2}, Hom(V_{1}, U_{1} ⊗ W ⊗ U_{2})) \ar[d]^{q_{6}} \\ Hom(V_{1} ⊗ V_{2}, U_{1} ⊗ W ⊗ U_{2}) \ar[r]^{Hom(s,Id_{U_{1}⊗W ⊗U_{2}} )} & Hom(V_{2} ⊗ V_{1}, U_{1} ⊗ W ⊗ U_{2})}

The q and s maps are invertible, and the commutativity around the outside of the diagram can be checked directly, using the formula for q^{−1} from Lemma 1.45. Starting in the upper left corner with U_{1}⊗Hom(V_{1}⊗V_{2}, W)⊗U_{2}, for A ∈ Hom(V_{1}⊗ V_{2}, W), u ∈ U_{1}, v ∈ U_{2}, y ∈ V_{1}, z ∈ V_{2},

u ⊗ A ⊗ v ↦ q_{6}(n_{10}∘[Id_{U_{1}} ⊗ n_{4}] ∘ (n_{8}([Id_{U_{1}} ⊗ n_{3}](u ⊗ (q_{2}^{−1}(A ∘ s)) ⊗ v)))) :
z ⊗ y ↦ ((n_{10}∘[Id_{U_{1}} ⊗ n_{4}] ∘ (n_{8}([Id_{U_{1}} ⊗ n_{3}](u ⊗ (q_{2}^{−1}(A ∘ s)) ⊗ v))))(z))(y)
= (n_{10}([Id_{U_{1}} ⊗ n_{4}](u ⊗ ((q_{2}^{−1}(A ∘ s))(z)) ⊗ v)))(y)
= u ⊗ (((q_{2}^{−1}(A ∘ s))(z))(y)) ⊗ v
= u ⊗ ((A ∘ s)(z ⊗ y)) ⊗ v = u ⊗ (A(y ⊗ z)) ⊗ v,
u ⊗ A ⊗ v ↦ (q_{5}(n_{9} ∘ [Id_{U_{1}} ⊗ n_{1}] ∘ (n_{7}([Id_{U_{1}} ⊗ n_{2}](u ⊗ (q_{1}^{−1}(A)) ⊗ v))))) ∘ s :
z ⊗ y ↦ ((n_{9} ∘ [Id_{U_{1}} ⊗ n_{1}] ∘ (n_{7}([Id_{U_{1}} ⊗ n_{2}](u ⊗ (q_{1}^{−1}(A)) ⊗ v))))(y))(z)
= (n_{9}([Id_{U_{1}} ⊗ n_{1}](u ⊗ v ⊗ ((q_{1}^{−1}(A))(y)))))(z)
= u ⊗ (((q_{1}^{−1}(A))(y))(z)) ⊗ v = u ⊗ (A(y ⊗ z)) ⊗ v.

The following Lemma 4.9 shows how T_{V_{1},V_{2};W} is related to some switching maps, which are involutions in the case V_{1} = V_{2}; an analogue for the scalar case T_{V} is Theorem 3.115.

Lemma 4.9. For any V_{1}, V_{2}, W, the following diagram is commutative. If V_{1} and V_{2} are both finite-dimensional, then all the arrows are invertible.

\xymatrix{V_{1}^{∗} ⊗ V_{2}^{∗} ⊗ W \ar[rr]^{[s_{2}⊗Id_{W}]} \ar[d]_{[Id_{V ^{∗}_{1}} ⊗k_{V_{2}W}]} \ar[ddr]^{k_{V_{1},V ^{∗}_{2}⊗W}} &  & V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ W \ar[d]^{k_{V_{2},V ^{∗}_{1}⊗W}} \ar[dl]_{[Id_{V ^{∗}_{2}} ⊗k_{V_{1}W}]} \\ V_{1}^{∗} ⊗ Hom(V_{2}, W) \ar[dd]^{k_{V_{1},Hom(V_{2},W)}} & V_{2}^{∗} ⊗ Hom(V_{1}, W) \ar[d]_{n} \ar[ddr]_{k_{V_{2},Hom(V_{1},W)}} & Hom(V_{2}, V_{1}^{∗} ⊗ W) \ar[dd]_{Hom(Id_{V_{2}} ,k_{V_{1}W})} \\  & Hom(V_{1}, V_{2}^{∗} ⊗ W) \ar[dl]^{Hom(Id_{V_{1}} ,k_{V_{2}W})} \\ Hom(V_{1}, Hom(V_{2}, W)) \ar[rr]_{T_{V_{1},V_{2};W}} \ar[d]_{q_{1}} &  & Hom(V_{2}, Hom(V_{1}, W)) \ar[d]^{q_{2}} \\ Hom(V_{1} ⊗ V_{2}, W) \ar[rr]^{Hom(s_{1},Id_{W})} &  & Hom(V_{2} ⊗ V_{1}, W) \\ (V_{1} ⊗ V_{2})^{∗} ⊗ W \ar[u]^{k_{V_{1}⊗V_{2},W}} \ar[rr]^{[s_{1}^{∗}⊗Id_{W}]} &  & (V_{2} ⊗ V_{1})^{∗} ⊗ W \ar[u]_{k_{V_{2}⊗V_{1},W}} \\ Hom(V_{1} ⊗ V_{2}, 𝕂 ⊗ 𝕂) ⊗ W \ar[u]_{[Hom(Id_{V_{1}⊗V_{2}} ,l)⊗Id_{W}]} &  & Hom(V_{2} ⊗ V_{1}, 𝕂 ⊗ 𝕂) ⊗ W \ar[u]^{[Hom(Id_{V_{2}⊗V_{1}} ,l)⊗Id_{W}]} \\ V_{1}^{∗} ⊗ V_{2}^{∗} ⊗ W \ar[u]^{[j_{1}⊗Id_{W}]} \ar[rr]^{[s_{2}⊗Id_{W}]} &  & V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ W \ar[u]_{[j_{2}⊗Id_{W}]}}

Proof. The block with the q maps is exactly Lemma 4.1, and the next lower block is commutative by Lemma 1.58. The lowest block is easy to check, where the abbreviation j̃_{1} = Hom(Id_{V_{1}⊗V_{2}}, l) ∘ j_{1} as in Notation 2.43 could be used, so that the vertical composite is [j̃_{1} ⊗ Id_{W}]. Inside the upper rectangle, the blocks on the left and right are commutative by Lemma 1.58 again, and its upper block is commutative by a variation on Lemma 1.61. The lower block in the upper rectangle is commutative: using Definition 4.2 and ϕ⊗A ∈ V_{2}^{∗} ⊗Hom(V_{1}, W), u ∈ V_{1}, v ∈ V_{2},

T_{V_{1},V_{2};W} ∘ Hom(Id_{V_{1}}, k_{V_{2},W}) ∘ n :
ϕ ⊗ A ↦ (t_{V_{1},Hom(V_{2},W)}^{W}(k_{V_{2},W} ∘ (n(ϕ ⊗ A)))) ∘ d_{V_{2},W} :
v ↦ (d_{V_{2},W} (v)) ∘ k_{V_{2},W} ∘ (n(ϕ ⊗ A)) :
u ↦ (k_{V_{2},W} (ϕ ⊗ (A(u))))(v)
= ϕ(v) · (A(u)) = ((k_{V_{2},Hom(V_{1},W)}(ϕ ⊗ A))(v))(u).

So the top rectangle is commutative.

Lemma 4.10. For any V_{1}, V_{2}, V_{3}, W, if V_{1} and V_{2} are finite-dimensional, then all the arrows in the following diagram are invertible and the diagram is commutative.

\xymatrix{Hom(V_{1}, Hom(V_{2}, W)) ⊗ V_{3} \ar[r]^{[T_{V_{1},V_{2};W} ⊗Id_{V_{3}} ]} \ar[d]_{n_{1}} & Hom(V_{2}, Hom(V_{1}, W)) ⊗ V_{3} \ar[d]^{n_{2}} \\ Hom(V_{1}, V_{3} ⊗ Hom(V_{2}, W)) & Hom(V_{2}, Hom(V_{1}, W) ⊗ V_{3}) \\ Hom(V_{1}, V_{3} ⊗ V_{2}^{∗} ⊗ W) \ar[u]_{Hom(Id_{V_{1}} ,[Id_{V_{3}} ⊗k_{V_{2}W}])} & Hom(V_{2}, V_{1}^{∗} ⊗ W ⊗ V_{3}) \ar[u]_{Hom(Id_{V_{2}} ,[k_{V_{1}W} ⊗Id_{V_{3}} ])} \\ V_{1}^{∗} ⊗ V_{3} ⊗ V_{2}^{∗} ⊗ W \ar[u]_{k_{V_{1},V_{3}⊗V ^{∗}_{2}⊗W}} \ar[r]^{s_{1}} & V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ W ⊗ V_{3} \ar[u]_{k_{V_{2},V ^{∗}_{1}⊗W ⊗V_{3}}}}

Proof. The following diagram is commutative, where the column on the left matches the left column in the above diagram, and the column on the right uses two maps from the diagram in Lemma 4.9.

\xymatrix{Hom(V_{1}, V_{3} ⊗ Hom(V_{2}, W)) & Hom(V_{1}, Hom(V_{2}, W)) ⊗ V_{3} \ar[l]^{n_{1}} \\ Hom(V_{1}, V_{3} ⊗ V_{2}^{∗} ⊗ W) \ar[u]^{Hom(Id_{V_{1}} ,[Id_{V_{3}} ⊗k_{V_{2}W}])} & Hom(V_{1}, V_{2}^{∗} ⊗ W) ⊗ V_{3} \ar[u]_{[Hom(Id_{V_{1}} ,k_{V_{2}W})⊗Id_{V_{3}} ]} \ar[l]^{n_{3}} \\ V_{1}^{∗} ⊗ V_{3} ⊗ V_{2}^{∗} ⊗ W \ar[u]^{k_{V_{1},V_{3}⊗V ^{∗}_{2}⊗W}} & V_{1}^{∗} ⊗ V_{2}^{∗} ⊗ W ⊗ V_{3} \ar[u]_{[k_{V_{1},V ^{∗}_{2}⊗W} ⊗Id_{V_{3}} ]} \ar[l]^{[Id_{V ^{∗}_{1}} ⊗s]}}

The upper square is commutative by Lemma 1.41, and the lower square by a variation on Lemma 1.61. The above diagram is abbreviated to appear in the left block of the following diagram, by labeling its four corners (M_{11}, M_{12}, M_{21}, M_{22}), and its upward vertical composites as the vertical arrows a_{1}, [a_{2} ⊗ Id_{V_{3}}].

\xymatrix{M_{11} & M_{12} \ar[l]_{n_{1}} \ar[r]^{[T_{V_{1},V_{2};W} ⊗Id_{V_{3}} ]} & M_{13} \ar[r]^{n_{2}} & Hom(V_{2}, Hom(V_{1}, W) ⊗ V_{3}) \\  &  &  & Hom(V_{2}, V_{1}^{∗} ⊗ W ⊗ V_{3}) \ar[u]^{Hom(Id_{V_{2}} ,[k_{V_{1}W} ⊗Id_{V_{3}} ])} \\ M_{21} \ar[uu]^{a_{1}} & M_{22} \ar[uu]^{[a_{2}⊗Id_{V_{3}} ]} \ar[l]^{[Id_{V ^{∗}_{1}} ⊗s]} \ar[r]^{[[s_{2}⊗Id_{W}]⊗Id_{V_{3}} ]} & M_{23} \ar[uu]^{[a_{3}⊗Id_{V_{3}} ]} \ar[ur]_{k_{V_{2},V ^{∗}_{1}⊗W ⊗V_{3}}}}

The other two spaces are similarly related to the right column in the top square from Lemma 4.9:

M_{13} = Hom(V_{2}, Hom(V_{1}, W)) ⊗ V_{3}
M_{23} = V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ W ⊗ V_{3},
a_{3} = Hom(Id_{V_{2}}, k_{V_{1}W}) ∘ k_{V_{2},V_{1}^{∗}⊗W} ,

so that the commutativity of the middle block follows from the commutativity of the top block from Lemma 4.9, together with Lemma 1.36. The right block is a mirror image analogue of the left block, but without the switching map, so the verification that it is commutative again uses Lemma 1.41 and Lemma 1.61. Letting s_{1} = [[s_{2} ⊗ Id_{W}] ⊗ Id_{V_{3}}] ∘ [Id_{V_{1}^{∗}} ⊗ s^{−1}], the commutativity around the outside of the diagram gives the claim of the Lemma.

Exercise 4.11. If V_{1}, V_{2}, W have metrics g_{1}, g_{2}, h, then
T_{V_{1},V_{2};W} : Hom(V_{1}, Hom(V_{2}, W)) → Hom(V_{2}, Hom(V_{1}, W))

is an isometry with respect to the induced metrics.

Hint. From Lemma 4.1, T_{V_{1},V_{2};W} = q_{2}^{−1} ∘ Hom(s, Id_{W}) ∘ q_{1} is a composite of isometries, using Theorem 3.45, Exercise 3.66, and Exercise 3.77. In the special case V_{1} = V_{2}, g_{1} = g_{2}, Exercise 3.122 applies to the involution T_{V_{1};W}. A further special case is W = 𝕂, so Corollary 3.48 on the involution T_{V} follows from this claim and Exercise 3.74.

The transpose for bilinear forms can be applied to vector valued \underline{trilinear} forms: elements of Hom(X, Hom(Y, Hom(Z, U))), to switch the first and second, or second and third, inputs in expressions such as ((h(x))(y))(z) ∈ U. An operator switching the first and third inputs can be expressed in terms of T maps, or in terms of q and s maps in analogy with Lemma 4.1, but in more than one way, as shown by the following Lemma 4.12 and Equation (4.5).

Lemma 4.12. For any vector spaces X, Y, Z, U, the following diagram is commutative.

\xymatrix{ & Hom(X, Hom(Y, Hom(Z, U))) \ar[dl]_{Hom(Id_{X},T_{Y,Z;U})} \ar[dr]^{T_{X,Y ;Hom(Z,U)}} \\ Hom(X, Hom(Z, Hom(Y, U))) \ar[d]^{T_{X,Z;Hom(Y,U)}} &  & Hom(Y,Hom(X, Hom(Z, U))) \ar[d]_{Hom(Id_{Y} ,T_{X,Z;U})} \\ Hom(Z, Hom(X, Hom(Y, U))) \ar[dr]_{Hom(Id_{Z},T_{X,Y ;U})} &  & Hom(Y,Hom(Z, Hom(X, U))) \ar[dl]^{T_{Y,Z;Hom(X,U)}} \\  & Hom(Z, Hom(Y, Hom(X, U)))}

Proof. The claim follows from finding a map

a_{1} : Hom(X, Hom(Y, Hom(Z, U))) → Hom(Z, Hom(Y, Hom(X, U))),

equal to both downward composites on the left and right sides of the above diagram. This first diagram corresponds to the right side.

\xymatrix{Hom(X,Hom(Y,Hom(Z,U))) \ar@{-->}[drr]^{T_{X,Y ;Hom(Z,U)}} \ar[ddr]_{q_{1}} \\  &  & Hom(Y,Hom(X, Hom(Z,U))) \ar[ddddddd]_{Hom(Id_{Y} ,T_{X,Z;U})} \ar[ddl]^{q_{2}} \ar[dddl]^{Hom(Id_{Y} ,q_{5})} \\  & Hom(X ⊗Y, Hom(Z,U)) \ar[dl]_{q_{3}} \ar[d]_{Hom(s_{1},Id_{Hom(Z,U)})} \\ Hom(X ⊗Y ⊗Z, U) \ar[d]^{Hom([s_{1}⊗Id_{Z}],Id_{U})} & Hom(Y ⊗X, Hom(Z,U)) \ar[dl]^{q′_{3}} \\ Hom(Y ⊗X ⊗Z, U) \ar[d]^{Hom([Id_{Y} ⊗s_{2}],Id_{U})} & Hom(Y, Hom(X ⊗Z, U)) \ar[d]^{Hom(Id_{Y} ,Hom(s_{2},Id_{U}))} \ar[l]_{q_{4}} \\ Hom(Y ⊗Z⊗X, U) \ar[d]^{Hom([s_{3}⊗Id_{X}],Id_{U})} & Hom(Y, Hom(Z⊗X), U) \ar[l]^{q′_{4}} \\ Hom(Z⊗Y ⊗X, U) & Hom(Y ⊗Z, Hom(X, U)) \ar[ul]_{q′′_{3}} \ar[d]_{Hom(s_{3},Id_{Hom(X,U)})} \\  & Hom(Z⊗Y, Hom(X, U)) \ar[ul]^{q′′′_{3}} \\  &  & Hom(Y,Hom(Z, Hom(X,U))) \ar@{-->}[dll]^{T_{Y,Z;Hom(X,U)}} \ar[uul]_{q′_{2}} \ar[uuul]_{Hom(Id_{Y} ,q_{6})} \\ Hom(Z,Hom(Y,Hom(X,U))) \ar[uur]^{q′_{1}}}

All the arrows are invertible — the q maps convert the transpose maps to switching maps. The right blocks are commutative by Lemma 4.1 and Definition 4.2, with the right center block also using Lemma 1.6. The three left blocks are commutative by Lemma 1.47 and the two center blocks by Lemma 1.48.

This second diagram, corresponding to the left side of the claim, is not exactly a mirror image of the first but is commutative in the same way, using Lemma 4.1, Lemma 1.6, Lemma 1.47, and Lemma 1.48.

\xymatrix{ &  & Hom(X,Hom(Y,Hom(Z,U))) \ar@{-->}[dll]_{Hom(Id_{X},T_{Y,Z;U})} \ar[ddl]^{Hom(Id_{X},q_{7})} \\ Hom(X, Hom(Z, Hom(Y,U))) \ar[ddddddd]^{T_{X,Z;Hom(Y,U)}} \ar[ddr]^{Hom(Id_{X},q_{8})} \ar[dddr]_{q_{11}} \\  & Hom(X, Hom(Y ⊗Z,U)) \ar[dr]^{q_{9}} \ar[d]^{Hom(Id_{X},Hom(s_{3},Id_{U}))} \\  & Hom(X, Hom(Z⊗Y,U)) \ar[dr]_{q′_{9}} & Hom(X ⊗Y ⊗Z, U) \ar[d]_{Hom([Id_{X}⊗s_{3}],Id_{U})} \\  & Hom(X ⊗Z, Hom(Y, U)) \ar[d]_{Hom(s_{2},Id_{Hom(Y,U)})} \ar[r]^{q_{10}} & Hom(X ⊗Z⊗Y, U) \ar[d]_{Hom([s_{2}⊗Id_{Y} ],Id_{U})} \\  & Hom(Z⊗X, Hom(Y, U)) \ar[r]_{q′_{10}} & Hom(Z⊗X ⊗Y, U) \ar[d]_{Hom([Id_{Z}⊗s_{1}],Id_{U})} \\  & Hom(Z, Hom(X ⊗Y, U)) \ar[ur]^{q′′_{9}} \ar[d]^{Hom(Id_{Z},Hom(s_{1},Id_{U}))} & Hom(Z⊗Y ⊗X, U) \\  & Hom(Z, Hom(Y ⊗X, U)) \ar[ur]_{q′′′_{9}} \\ Hom(Z, Hom(X, Hom(Y,U))) \ar@{-->}[drr]_{Hom(Id_{Z},T_{X,Y ;U})} \ar[uur]_{Hom(Id_{Z},q′_{8})} \ar[uuur]^{q_{12}} \\  &  & Hom(Z,Hom(Y,Hom(X,U))) \ar[uul]_{Hom(Id_{Z},q′_{7})}}

The composites of permutations in the diagrams are equal to the same switching map s_{4}:

[s_{3} ⊗ Id_{X}] ∘ [Id_{Y} ⊗ s_{2}] ∘ [s_{1} ⊗ Id_{Z}] = [Id_{Z} ⊗ s_{1}] ∘ [s_{2} ⊗ Id_{Y}] ∘ [Id_{X} ⊗ s_{3}]
= s_{4} : X ⊗ Y ⊗ Z → Z ⊗ Y ⊗ X.

By Lemma 1.48, q_{3} ∘ q_{1} = q_{9} ∘ Hom(Id_{X}, q_{7}) and q′′′_{3} ∘ q′_{1} = q′′′_{9} ∘ Hom(Id_{Z} ∘ q′_{7}). So, the two diagrams fit together as claimed, with the downward composite in the left column of the first diagram being equal to the composite in the right column of the second, giving the required map

(4.5)a_{1} = (q′′′_{3}∘ q′_{1})^{−1} ∘ Hom([s_{3} ⊗ Id_{X}]∘[Id_{Y} ⊗ s_{2}]∘[s_{1}⊗ Id_{Z}], Id_{U}) ∘ q_{3} ∘ q_{1}
= Hom(Id_{Z}, (q′_{7})^{−1}) ∘ (q′′′_{9})^{−1} ∘ Hom(s_{4}, Id_{U}) ∘ q_{9} ∘ Hom(Id_{X}, q_{7}).

4.2. Symmetric bilinear forms

Definition 4.13. A W -valued form h ∈ Hom(V, Hom(V, W)) is \underline{symmetric} means: h = T_{V;W} (h). h is \underline{antisymmetric} means: h = −T_{V;W} (h). Let Sym(V; W) denote the subspace of symmetric forms, and Alt(V; W) the subspace of antisymmetric forms.

It follows from Lemma 1.114 and Lemma 4.4 that if \frac{1}{2} ∈ 𝕂, then T_{V;W} produces a direct sum

(4.6)  Hom(V, Hom(V, W)) = Sym(V; W) ⊕ Alt(V; W).

Remark 4.14. The direct sum (4.6) is canonical, and so is the decomposition of any form h into its symmetric and antisymmetric parts

(4.7)  \frac{1}{2}(h + T_{V;W} (h)) + \frac{1}{2}(h − T_{V;W} (h)).

However, there are several other involutions appearing in the V = V_{1} = V_{2} case of Lemma 4.9, and some of the other spaces admit distinct but equivalent direct sums as in Example 1.138. Recalling the direct sum V ⊗ V = S^{2}V ⊕ Λ^{2}V produced by the involution s as in Example 1.118, Example 1.139 applies to the involutions T_{V;W} and Hom(s, Id_{W}) from Lemma 4.1 and Lemma 4.9, so the map

q : Hom(V, Hom(V, W)) → Hom(V ⊗ V, W)

respects both of the direct sums on the target:

Hom(V ⊗ V, W) = {A : A ∘ s = A} ⊕ {A : A ∘ s = −A},
Hom(V ⊗ V, W) = Hom(S^{2}V, W) ⊕ Hom(Λ^{2}V, W).

Example 4.15. It follows from Lemma 4.5 that for a map of the form

h = (Hom(Id_{V} , k_{V W}) ∘ n_{1})(g ⊗ w),

with g : V → V^{∗}, w ∈ W, if g is symmetric, or antisymmetric, then so is h.

Definition 4.16. For any U, V, W, the \underline{pullback} of a W -valued form h : V → Hom(V, W) by a map H : U → V is another W -valued form Hom(H, Id_{W}) ∘ h ∘ H : U → Hom(U, W).

In the case W = 𝕂, this coincides with the previously defined pullback (Definition 3.7).

Lemma 4.17. For maps H : U → V, G : W_{1} → W_{2}, and a form h : V → Hom(V, W_{1}),

T_{U;W_{2}}(Hom(H, G) ∘ h ∘ H) = Hom(H, G) ∘ (T_{V;W_{1}}(h)) ∘ H.

The map

Hom(H, Hom(H, G)) : Hom(V, Hom(V, W_{1})) → Hom(U, Hom(U, W_{2}))

respects the direct sums Sym(V; W_{1}) ⊕ Alt(V; W_{1}) → Sym(U; W_{2}) ⊕ Alt(U; W_{2}).

Proof. The first claim is a special case of Lemma 4.6. The claim about the direct sums follows from Lemma 1.120 and Lemma 4.4.

The G = Id_{W} case of Lemma 4.17 shows that the pullback by H : U → V of a symmetric form h : V → Hom(V, W) is a symmetric form U → Hom(U, W), and similarly, the pullback of an antisymmetric form is antisymmetric.

Notation 4.18. For h_{1} : V_{1} → Hom(V_{1}, W), and h_{2} : V_{2} → Hom(V_{2}, W), and a direct sum V = V_{1} ⊕ V_{2}, let h_{1} ⊕ h_{2} : V → Hom(V, W) denote the form

Hom(P_{1}, Id_{W}) ∘ h_{1} ∘ P_{1} + Hom(P_{2}, Id_{W}) ∘ h_{2} ∘ P_{2}.

In the W = 𝕂 case, this is exactly the construction of Notation 3.9.

Lemma 4.19. T_{V;W} (h_{1} ⊕ h_{2}) = (T_{V_{1};W} (h_{1})) ⊕ (T_{V_{2};W} (h_{2})).

Proof. The proof proceeds exactly as in Theorem 3.10, using Lemma 1.14 and Lemma 1.6.

It follows that the direct sum of symmetric W -valued forms is symmetric, and similarly, the direct sum of antisymmetric forms is antisymmetric.

Working with the tensor product of vector valued forms is simpler than the scalar case (Notation 3.13), since the scalar multiplication is omitted. If h_{1} : V_{1} → Hom(V_{1}, W_{1}) and h_{2} : V_{2} → Hom(V_{2}, W_{2}) are two vector valued forms, then the map

j ∘ [h_{1} ⊗ h_{2}] : V_{1} ⊗ V_{2} → Hom(V_{1} ⊗ V_{2}, W_{1} ⊗ W_{2})

has output

((j ∘ [h_{1} ⊗ h_{2}])(u_{1} ⊗ u_{2}))(v_{1} ⊗ v_{2}) = ((h_{1}(u_{1}))(v_{1})) ⊗ ((h_{2}(u_{2}))(v_{2})) ∈ W_{1} ⊗ W_{2},

so it is a W_{1} ⊗ W_{2}-valued form.

Theorem 4.20.

T_{V_{1}⊗V_{2};W_{1}⊗W_{2}}(j ∘ [h_{1} ⊗ h_{2}]) = j ∘ [(T_{V_{1};W_{1}}(h_{1})) ⊗ (T_{V_{2};W_{2}}(h_{2}))].

Proof. In analogy with the proof of Theorem 3.12, the following diagram is commutative:

\xymatrix{V_{1} ⊗ V_{2} \ar[d]^{d_{V_{1}⊗V_{2},W_{1}⊗W_{2}}} \ar[r]^{[d_{V_{1}W_{1}} ⊗d_{V_{2}W_{2}} ]} & M_{1} \ar[d]^{j′} \\ Hom(Hom(V_{1}⊗V_{2},W_{1}⊗W_{2}),W_{1}⊗W_{2}) \ar[r]_{Hom(j,Id_{W_{1}⊗W_{2}} )} & M_{2}}

where

M_{1} = Hom(Hom(V_{1}, W_{1}), W_{1}) ⊗ Hom(Hom(V_{2}, W_{2}), W_{2})
M_{2} = Hom(Hom(V_{1}, W_{1}) ⊗ Hom(V_{2}, W_{2}), W_{1} ⊗ W_{2});
v_{1} ⊗ v_{2} ↦ (Hom(j, Id_{W_{1}⊗W_{2}}) ∘ d_{V_{1}⊗V_{2},W_{1}⊗W_{2}})(v_{1} ⊗ v_{2})
= (d_{V_{1}⊗V_{2},W_{1}⊗W_{2}}(v_{1} ⊗ v_{2})) ∘ j :
A ⊗ B ↦ [A ⊗ B](v_{1} ⊗ v_{2}) = (A(v_{1})) ⊗ (B(v_{2})),
v_{1} ⊗ v_{2} ↦ (j′ ∘ [d_{V_{1}W_{1}} ⊗ d_{V_{2}W_{2}}])(v_{1} ⊗ v_{2})
= j′((d_{V_{1}W_{1}}(v_{1})) ⊗ (d_{V_{2}W_{2}}(v_{2}))) :
A ⊗ B ↦ (A(v_{1})) ⊗ (B(v_{2})).

The statement of the Theorem follows from Lemma 1.6, the above diagram, Lemma 1.37, and Lemma 1.36:

LHS = (t_{V_{1}⊗V_{2},Hom(V_{1}⊗V_{2},W_{1}⊗W_{2})}^{W_{1}⊗W_{2}}(j ∘ [h_{1} ⊗ h_{2}])) ∘ d_{V_{1}⊗V_{2},W_{1}⊗W_{2}}
= Hom([h_{1} ⊗ h_{2}], Id_{W_{1}⊗W_{2}}) ∘ Hom(j, Id_{W_{1}⊗W_{2}}) ∘ d_{V_{1}⊗V_{2},W_{1}⊗W_{2}}
= Hom([h_{1} ⊗ h_{2}], Id_{W_{1}⊗W_{2}}) ∘ j′ ∘ [d_{V_{1}W_{1}} ⊗ d_{V_{2}W_{2}}]
= j ∘ [Hom(h_{1}, Id_{W_{1}}) ⊗ Hom(h_{2}, Id_{W_{2}})] ∘ [d_{V_{1}W_{1}} ⊗ d_{V_{2}W_{2}}]
= j ∘ [(T_{V_{1};W_{1}}(h_{1})) ⊗ (T_{V_{2};W_{2}}(h_{2}))].

It follows that the tensor product of symmetric forms is symmetric, as is the tensor product of antisymmetric forms.

In the W_{1} = 𝕂 case, the tensor product of a scalar valued form h_{1} : V_{1} → V_{1}^{∗} and a vector valued form h_{2} : V_{2} → Hom(V_{2}, W) is a form j ∘ [h_{1} ⊗ h_{2}] with values in 𝕂 ⊗ W. The map Hom(Id_{V_{1}⊗V_{2}}, l_{W}) ∘ j ∘ [h_{1} ⊗ h_{2}] is a W -valued form.

Corollary 4.21. For h_{1} : V_{1} → V_{1}^{∗} and h_{2} : V_{2} → Hom(V_{2}, W), the following W -valued forms are equal.

T_{V_{1}⊗V_{2};W} (Hom(Id_{V_{1}⊗V_{2}}, l_{W}) ∘ j ∘ [h_{1} ⊗ h_{2}])
= Hom(Id_{V_{1}⊗V_{2}}, l_{W}) ∘ j ∘ [(T_{V_{1}}(h_{1})) ⊗ (T_{V_{2};W} (h_{2}))].

Proof. The equality follows immediately from Lemma 4.6, the previous Theorem, and the equality T_{V;𝕂} = T_{V}.

Exercise 4.22. Let V = U ⊕ Hom(U, L) be a direct sum with operators P_{i}, Q_{i}. Then,

(4.8)  Hom(P_{1}, Id_{L}) ∘ P_{2} + Hom(P_{2}, Id_{L}) ∘ d_{UL} ∘ P_{1}

is a symmetric L-valued form on V. If d_{UL} is invertible, then this form is also invertible.

Hint. The proof that the form (4.8) is symmetric is the same as the calculation from Lemma 3.102, but using Lemma 1.14 and Lemma 1.17 in their full generality. If d_{UL} is invertible (for example, as in Proposition 3.78), then the inverse of the form is

Q_{1} ∘ d_{UL}^{−1} ∘ Hom(Q_{2}, Id_{L}) + Q_{2} ∘ Hom(Q_{1}, Id_{L}).

Exercise 4.23. Let V = U ⊕ Hom(U, L) as in Exercise 4.22. Given maps E : W → U and h : U → Hom(W, L), the following are equivalent.

(1) The L-valued form h ∘ E : W → Hom(W, L) is antisymmetric.

(2) The pullback of the symmetric L-valued form (4.8) by the map

Q_{1} ∘ E + Q_{2} ∘ Hom(h, Id_{L}) ∘ d_{W L} : W → V

is 0_{Hom(W,Hom(W,L))}.

Hint. The statement is analogous to Exercise 3.109, but uses the generalized notion of pullback from Definition 4.16.

Exercise 4.24. Suppose V = V_{1} ⊕ V_{2}, and there is an invertible map g : V → Hom(V, L) so that these pullbacks are zero:

Hom(Q_{1}, Id_{L}) ∘ g ∘ Q_{1} = 0_{Hom(V_{1},Hom(V_{1},L))},
Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{2} = 0_{Hom(V_{2},Hom(V_{2},L))}.

Then these maps are invertible:

Hom(Q_{1}, Id_{L}) ∘ g ∘ Q_{2} : V_{2} → Hom(V_{1}, L),
Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{1} : V_{1} → Hom(V_{2}, L).
Hint. The inverses are P_{2} ∘ g^{−1} ∘ Hom(P_{1}, Id_{L}), P_{1} ∘ g^{−1} ∘ Hom(P_{2}, Id_{L}).

Exercise 4.25. ([EPW]) Let V = V_{1} ⊕V_{2} and g : V → Hom(V, L) be as in the previous Exercise. Then there is another direct sum V = V_{1} ⊕ Hom(V_{1}, L), defined by operators P_{1}, Q_{1}, and

P′_{2} = Hom(Q_{1}, Id_{L}) ∘ g : V → Hom(V_{1}, L),
Q′_{2} = g^{−1} ∘ Hom(P_{1}, Id_{L}) : Hom(V_{1}, L) → V.

If, also, g is symmetric, then g is equal to the L-valued form induced by this direct sum, as in Exercise 4.22:

g = Hom(P_{1}, Id_{L}) ∘ P′_{2} + Hom(P′_{2}, Id_{L}) ∘ d_{V_{1},L} ∘ P_{1}.
Hint. It is easy to check that P′_{2} ∘ Q_{1} is zero, and P′_{2} ∘ Q′_{2} is the identity.
Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{1} ∘ P_{1} ∘ Q′_{2}
= Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{1} ∘ P_{1} ∘ g^{−1} ∘ Hom(P_{1} ∘ Id_{L})
= Hom(Q_{2}, Id_{L}) ∘ g ∘ (Id_{V} − Q_{2} ∘ P_{2}) ∘ g^{−1} ∘ Hom(P_{1}, Id_{L})
= 0_{Hom(Hom(V_{1},L),Hom(V_{2},L))}.

By the previous Exercise, Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{1} is invertible, so

P_{1} ∘ Q′_{2} = 0_{Hom(Hom(V_{1},L),V_{1})}.
Q_{1} ∘ P_{1} + Q′_{2} ∘ P′_{2}
= Q_{1} ∘ P_{1} + (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ Q′_{2} ∘ P′_{2} ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2})
= Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2} ∘ g^{−1} ∘ Hom(P_{1}, Id_{L}) ∘ Hom(Q_{1}, Id_{L}) ∘ g ∘ Q_{2} ∘ P_{2}
= Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2} = Id_{V} ,

using the inverse formula from the previous Exercise. As for the claimed equality,

RHS = Hom(P_{1}, Id_{L}) ∘ Hom(Q_{1}, Id_{L}) ∘ g

+Hom(Hom(Q_{1}, Id_{L}) ∘ g, Id_{L}) ∘ d_{V_{1},L} ∘ P_{1}

= Hom(P_{1}, Id_{L}) ∘ Hom(Q_{1}, Id_{L}) ∘ g ∘ (Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) +

Hom(g, Id_{L}) ∘ Hom(Hom(Q_{1}, Id_{L}), Id_{L}) ∘ d_{V_{1},L} ∘ P_{1}

= Hom(P_{1}, Id_{L}) ∘ Hom(Q_{1}, Id_{L}) ∘ g ∘ Q_{2} ∘ P_{2}

+Hom(P_{2}, Id_{L}) ∘ Hom(Q_{2}, Id_{L}) ∘ g ∘ Q_{2} ∘ P_{2} +Hom(g, Id_{L}) ∘ d_{V L} ∘ Q_{1} ∘ P_{1}

= Hom(Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}, Id_{L}) ∘ g ∘ Q_{2} ∘ P_{2} + (T_{V L}(g)) ∘ Q_{1} ∘ P_{1}
= g ∘ (Q_{2} ∘ P_{2} + Q_{1} ∘ P_{1}) = g.

4.3. Vector valued trace with respect to a metric

A metric g on V suggests that the scalar trace T r_{g} (Definition 3.32) can be generalized to a vector valued trace operator on W -valued forms, but at first there would appear to be two constructions of a map T r_{g;W} : Hom(V, Hom(V, W)) → W. One way would be to combine the previously constructed vector valued trace T r_{V^{∗};W} (Definition 2.50) and composition with g^{−1}, and another would be to start with the scalar trace T r_{g}, and tensor with Id_{W}. Of course, the two approaches have the same result.

Lemma 4.26. Given a metric g on V, the following diagram is commutative.

\xymatrix{Hom(V,Hom(V, W)) \ar[r]^{Hom(Id_{V},k_{V W}^{−1} )} \ar[d]_{Hom(g^{−1},k_{V W}^{−1} )} & Hom(V, V^{∗}⊗W) \ar[r]^{n_{1}^{−1}} \ar[dl]_{Hom(g^{−1},Id_{V ^{∗}⊗W})} & Hom(V, V^{∗})⊗W \ar[l]^{n_{1}} \ar[dl]_{[Hom(g^{−1},Id_{V ^{∗}})⊗Id_{W}]} \ar[ddl]^{[T r_{g}⊗Id_{W}]} \\ Hom(V^{∗}, V^{∗} ⊗ W) \ar[d]^{T r_{V ^{∗};W}} & Hom(V^{∗}, V^{∗}) ⊗W \ar[l]^{n} \ar[d]_{[T r_{V ^{∗}} ⊗Id_{W}]} \\ W & 𝕂 ⊗ W \ar[l]^{l_{W}}}

Proof. The upper left triangle is commutative by Lemma 1.6, and the upper right block is commutative by Lemma 1.41, with invertible n maps by the finitedimensionality of V. The lower square is the definition of T r_{V^{∗};W}, and the right triangle uses the definition of T r_{g}.

Definition 4.27. Given a metric g on V, an arbitrary vector space W, and a W -valued form h : V → Hom(V, W), the \underline{W -valued trace} with respect to g is the following element of W:

T r_{g;W} (h) = T r_{V^{∗};W} (k_{V W}^{−1} ∘ h ∘ g^{−1}).

Corollary 2.57 also gives the equality

T r_{V^{∗};W} (k_{V W}^{−1} ∘ h ∘ g^{−1}) = T r_{V;W} ([g^{−1} ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ h).

By the previous Lemma,

T r_{g;W} = T r_{V^{∗};W} ∘ Hom(g^{−1}, k_{V W}^{−1} ) = l_{W} ∘ [T r_{g} ⊗ Id_{W}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , k_{V W}^{−1} ).

Example 4.28. Given a metric g on V, if h is of the form h = (Hom(Id_{V} , k_{V W})∘ n_{1})(E ⊗ w), for E : V → V^{∗} and w ∈ W, then T r_{g;W} (h) = T r_{g}(E) · w, and if T r_{g}(E) = 0, then T r_{g;W} (h) = 0_{W}.

The previously defined scalar valued trace with respect to g (Definition 3.32) is exactly the W = 𝕂 case of the vector valued case:

Theorem 4.29. Given a metric g on V, for h : V → V^{∗}, T r_{g;𝕂}(h) = T r_{g}(h).

Proof. By Lemma 1.59, k_{V 𝕂} : V^{∗} ⊗ 𝕂 → V^{∗} is exactly the scalar multiplication appearing in Theorem 2.53, so that

T r_{g;𝕂}(h) = T r_{V^{∗};𝕂}(k_{V 𝕂}^{−1} ∘ h ∘ g^{−1}) = T r_{V^{∗}}(h ∘ g^{−1}) = T r_{g}(h).

Theorem 4.30. For any metric h^{ν} on 𝕂, as in Lemma 3.67, and a form h : 𝕂 → Hom(𝕂, W),

T r_{h^{ν};W} (h) = \frac{1}{ν} · (h(1))(1).

Proof.

T r_{h^{ν};W} (h) = T r_{𝕂;W} ([(h^{ν})^{−1} ⊗ Id_{W}] ∘ k_{𝕂W}^{−1} ∘ h)
= (l_{W} ∘ [(h^{ν})^{−1} ⊗ Id_{W}] ∘ k_{𝕂W}^{−1} ∘ h)(1)
= (\frac{1}{ν} · m^{−1})(h(1))
= \frac{1}{ν} (h(1))(1),

where the first step uses Corollary 2.56, and the last step uses the formula m^{−1} = d_{𝕂W} (1), from Definition 1.20. The intermediate step uses the commutativity of the diagram

\xymatrix{W & Hom(𝕂, W) \ar[l]^{m^{−1}} \\ 𝕂 ⊗ W \ar[u]^{ν·l_{W}} \ar[r]^{[h^{ν}⊗Id_{W}]} & 𝕂^{∗} ⊗ W \ar[u]^{k_{𝕂W}}}

λ ⊗ w ↦ (m^{−1} ∘ k_{𝕂W} ∘ [h^{ν} ⊗ Id_{W}])(λ ⊗ w)
= (k_{𝕂W} ((h^{ν}(λ)) ⊗ w))(1)
= ν · λ · 1 · w
= (ν · l_{W})(λ ⊗ w).

Theorem 4.31. Given a metric g on V, T r_{g;W} (T_{V;W} (h)) = T r_{g;W} (h).

Proof. Since V must be finite-dimensional, Lemma 4.5 and Lemma 4.26 apply.

T r_{g;W} ∘ T_{V;W} = l_{W} ∘ [T r_{g} ⊗ Id_{W}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , k_{V W}^{−1} )
∘Hom(Id_{V} , k_{V W}) ∘ n_{1} ∘ [T_{V} ⊗ Id_{W}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , k_{V W}^{−1} )
= l_{W} ∘ [T r_{g} ⊗ Id_{W}] ∘ [T_{V} ⊗ Id_{W}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , k_{V W}^{−1} )
= l_{W} ∘ [T r_{g} ⊗ Id_{W}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , k_{V W}^{−1} )
= T r_{g;W} ,

by Lemma 1.36 and Theorem 3.33, which stated that T r_{g} ∘ T_{V} = T r_{g}.

Corollary 4.32. Given a metric g on V, if h : V → Hom(V, W) is antisymmetric and \frac{1}{2} ∈ 𝕂, then T r_{g;W} (h) = 0_{W}.

Proposition 4.33. Given a metric g on V, the W -valued trace is invariant under pullback, that is, if H : U → V is invertible, then

T r_{H^{∗}∘g∘H;W} (Hom(H, Id_{W}) ∘ h ∘ H) = T r_{g;W} (h).

Proof. Using Corollary 2.57 and Lemma 1.58,

LHS = T r_{U^{∗};W} (k_{UW}^{−1} ∘ Hom(H, Id_{W}) ∘ h ∘ H ∘ H^{−1} ∘ g^{−1} ∘ (H^{∗})^{−1})
= T r_{V^{∗};W} ([(H^{∗})^{−1} ⊗ Id_{W}] ∘ k_{UW}^{−1} ∘ Hom(H, Id_{W}) ∘ h ∘ g^{−1})
= T r_{V^{∗};W} (k_{V W}^{−1} ∘ h ∘ g^{−1}) = RHS.

This statement and proof are analogous to Proposition 3.37.

Theorem 4.34. Given a metric g on V, for any map B : W → W′,

T r_{g;W′}(Hom(Id_{V} , B) ∘ h) = B(T r_{g;W} (h)).

Proof. Using Corollary 2.58 and Lemma 1.58,

LHS = T r_{V^{∗};W′}(k_{V W′}^{−1} ∘ Hom(Id_{V} , B) ∘ h ∘ g^{−1})
= T r_{V^{∗},W′}([Id_{V^{∗}} ⊗ B] ∘ k_{V W}^{−1} ∘ h ∘ g^{−1})
= B(T r_{V^{∗};W} (k_{V W}^{−1} ∘ h ∘ g^{−1})) = RHS.

Corollary 4.35. Given a metric g on V, and maps H : U → V, B : W → W′, if H is invertible then the following diagram is commutative.

\xymatrix{Hom(V, Hom(V, W)) \ar[r]^{T r_{g,W}} \ar[d]_{Hom(H,Hom(H,B))} & W \ar[d]^{B} \\ Hom(U, Hom(U, W′)) \ar[r]^{T r_{H^{∗}∘g∘H;W′}} & W′}

Proof. This follows from Proposition 4.33 and Theorem 4.34.

Proposition 4.36. Given metrics g_{1}, g_{2} on V_{1}, V_{2}, and a direct sum V = V_{1} ⊕ V_{2}, for W -valued forms h_{1} : V_{1} → Hom(V_{1}, W), h_{2} : V_{2} → Hom(V_{2}, W),

T r_{g_{1}⊕g_{2};W} (h_{1} ⊕ h_{2}) = T r_{g_{1};W} (h_{1}) + T r_{g_{2};W} (h_{2}) ∈ W.

Proof. First, Lemma 1.6 and Lemma 1.58 apply to simplify the following map from Hom(V_{I}, W) to V_{i}^{∗} ⊗ W:

[Q_{i}^{∗} ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ Hom(P_{I}, Id_{W})
= k_{V_{i}W}^{−1} ∘ Hom(Q_{i}, Id_{W}) ∘ Hom(P_{I}, Id_{W}) = k_{V_{i}W}^{−1} ∘ Hom(P_{I} ∘ Q_{i}, Id_{W})
= k_{V_{i}W}^{−1} , if i = I, or 0_{Hom(Hom(V_{I},W),V_{i}^{∗}⊗W)} if i ≠ I.

Then the formula (3.4) for (g_{1} ⊕ g_{2})^{−1} from Corollary 3.18 applies:

LHS = T r_{V;W} ([(g_{1} ⊕ g_{2})^{−1} ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ (h_{1} ⊕ h_{2}))
= T r_{V;W} ([(Q_{1} ∘ g_{1}^{−1} ∘ Q_{1}^{∗}) ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ Hom(P_{1}, Id_{W}) ∘ h_{1} ∘ P_{1}
+[(Q_{1} ∘ g_{1}^{−1} ∘ Q_{1}^{∗}) ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ Hom(P_{2}, Id_{W}) ∘ h_{2} ∘ P_{2}
+[(Q_{2} ∘ g_{2}^{−1} ∘ Q_{2}^{∗}) ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ Hom(P_{1}, Id_{W}) ∘ h_{1} ∘ P_{1}
+[(Q_{2} ∘ g_{2}^{−1} ∘ Q_{2}^{∗}) ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ Hom(P_{2}, Id_{W}) ∘ h_{2} ∘ P_{2})
= T r_{V;W} ([(Q_{1} ∘ g_{1}^{−1}) ⊗ Id_{W}] ∘ k_{V_{1}W}^{−1} ∘ h_{1} ∘ P_{1})
+T r_{V;W} ([(Q_{2} ∘ g_{2}^{−1}) ⊗ Id_{W}] ∘ k_{V_{2}W}^{−1} ∘ h_{2} ∘ P_{2})
= T r_{V_{1};W} ([(P_{1} ∘ Q_{1} ∘ g_{1}^{−1}) ⊗ Id_{W}] ∘ k_{V_{1}W}^{−1} ∘ h_{1})
+T r_{V_{2};W} ([(P_{2} ∘ Q_{2} ∘ g_{2}^{−1}) ⊗ Id_{W}] ∘ k_{V_{2}W}^{−1} ∘ h_{2}) = RHS.

The last steps used Corollary 2.57 and Lemma 1.36.

Theorem 4.37. For metrics g_{1}, g_{2} on V_{1}, V_{2}, and vector valued forms h_{1} : V_{1} → Hom(V_{1}, W_{1}), h_{2} : V_{2} → Hom(V_{2}, W_{2}),

T r_{{g_{1}⊗g_{2}};W_{1}⊗W_{2}}(j ∘ [h_{1} ⊗ h_{2}]) = (T r_{g_{1};W_{1}}(h_{1})) ⊗ (T r_{g_{2};W_{2}}(h_{2})) ∈ W_{1} ⊗ W_{2}.

Proof. The following diagram is commutative.

\xymatrix{V_{1} ⊗ V_{2} ⊗ W_{1} ⊗ W_{2} \ar[d]^{[[g_{1}⊗g_{2}]⊗Id_{W_{1}⊗W_{2}} ]} & V_{1} ⊗ W_{1} ⊗ V_{2} ⊗ W_{2} \ar[d]_{[[g_{1}⊗Id_{W_{1}} ]⊗[g_{2}⊗Id_{W_{2}} ]]} \ar[l]^{s_{1}} \\ V_{1}^{∗} ⊗ V_{2}^{∗} ⊗ W_{1} ⊗ W_{2} \ar[r]^{s} \ar[d]^{[j⊗Id_{W_{1}⊗W_{2}} ]} & V_{1}^{∗} ⊗ W_{1} ⊗ V_{2}^{∗} ⊗ W_{2} \ar[d]^{[k_{V_{1}W_{1}} ⊗k_{V_{2}W_{2}} ]} \\ Hom(V_{2} ⊗ V_{2}, 𝕂 ⊗ 𝕂) ⊗ W_{1} ⊗ W_{2} \ar[d]^{[Hom(Id_{V_{1}⊗V_{2}} ,l)⊗Id_{W_{1}⊗W_{2}} ]} & Hom(V_{1}, W_{1}) ⊗ Hom(V_{2}, W_{2}) \ar[d]^{j} \\ (V_{1} ⊗ V_{2})^{∗} ⊗ W_{1} ⊗ W_{2} \ar[r]^{k_{V_{1}⊗V_{2},W_{1}⊗W_{2}}} & Hom(V_{1} ⊗ V_{2}, W_{1} ⊗ W_{2})}

The commutativity of the lower part is exactly Lemma 2.32. The top square is easy to check, where the s_{1} map is as in Theorem 2.40. The statement of the Theorem follows from Corollary 2.63, using Lemma 1.36 and the formula for {g_{1} ⊗ g_{2}}^{−1} from Corollary 3.19:

LHS = T r_{V_{1}⊗V_{2};W_{1}⊗W_{2}}([([g_{1}^{−1} ⊗ g_{2}^{−1}] ∘ j^{−1} ∘ Hom(Id_{V_{1}⊗V_{2}}, l^{−1})) ⊗ Id_{W_{1}⊗W_{2}}]
∘k_{V_{1}⊗V_{2},W_{1}⊗W_{2}}^{−1} ∘ j ∘ [h_{1} ⊗ h_{2}])
= T r_{V_{1}⊗V_{2};W_{1}⊗W_{2}}(s_{1} ∘ [[g_{1}^{−1} ⊗ Id_{W_{1}}] ⊗ [g_{2}^{−1} ⊗ Id_{W_{2}}]]
∘[k_{V_{1}W_{1}}^{−1} ⊗ k_{V_{2}W_{2}}^{−1}] ∘ [h_{1} ⊗ h_{2}])
= T r_{V_{1}⊗V_{2};W_{1}⊗W_{2}}(s_{1} ∘ (j′_{3}(([g_{1}^{−1} ⊗ Id_{W_{1}}] ∘ k_{V_{1}W_{1}}^{−1} ∘ h_{1})
⊗([g_{2}^{−1} ⊗ Id_{W_{2}}] ∘ k_{V_{2}W_{2}}^{−1} ∘ h_{2}))))
= (T r_{V_{1};W_{1}}([g_{1}^{−1} ⊗ Id_{W_{1}}] ∘ k_{V_{1}W_{1}}^{−1} ∘ h_{1}))
⊗(T r_{V_{2};W_{2}}([g_{2}^{−1} ⊗ Id_{W_{2}}] ∘ k_{V_{2}W_{2}}^{−1} ∘ h_{2})) = RHS.

Corollary 4.38. For metrics g_{1}, g_{2} on V_{1}, V_{2}, a scalar valued form h_{1} : V_{1} → V_{1}^{∗}, and a W -valued form h_{2} : V_{2} → Hom(V_{2}, W),

T r_{{g_{1}⊗g_{2}};W} (Hom(Id_{V_{1}⊗V_{2}}, l_{W}) ∘ j ∘ [h_{1} ⊗ h_{2}]) = T r_{g_{1}}(h_{1}) · T r_{g_{2};W} (h_{2}).

Proof. Using Theorem 4.34, the previous Theorem, and Theorem 4.29,

LHS = l_{W} (T r_{{g_{1}⊗g_{2}};𝕂⊗W} (j ∘ [h_{1} ⊗ h_{2}]))
= l_{W} ((T r_{g_{1};𝕂}(h_{1})) ⊗ (T r_{g_{2};W} (h_{2}))) = RHS.

Theorem 4.39. If \frac{1}{2} ∈ 𝕂, and g and y are metrics on V and W, then the direct sum Sym(V; W) ⊕ Alt(V; W) is orthogonal with respect to the induced metric.

Proof. Since V is finite-dimensional, all the arrows in the diagram for Lemma 4.5 are invertible. Let H = Hom(Id_{V} , k_{V W}) ∘ n_{1}, so H and H^{−1} are isometries by Theorem 3.41, Theorem 3.45, and Lemma 3.73. Also, [T_{V} ⊗ Id_{W}] is an isometry by Corollary 3.48 and Theorem 3.28, so by Lemma 4.5 and Definition 4.2, T_{V;W} is an isometry, and an involution by Lemma 4.4. Then Lemma 3.55 applies to the direct sum produced by T_{V;W}.

By Theorem 3.56 and Theorem 3.60, Hom(V, V^{∗}) ⊗ W = (Sym(V) ⊗ W) ⊕ (Alt(V) ⊗ W) is an orthogonal direct sum. Since H respects the direct sums, by Lemma 4.5 and Lemma 1.120, it follows from Theorem 3.61 that the maps between Sym(V) ⊗ W and Sym(V; W), and between Alt(V) ⊗ W and Alt(V; W), are isometries.

Theorem 4.40. If T r_{V} (Id_{V}) ≠ 0, and g is a metric on V, then there is a direct sum Hom(V, Hom(V, W)) = W ⊕ ker(T r_{g;W}). If y is a metric on W, then the direct sum is orthogonal with respect to the induced metric.

Proof. By Theorem 3.53, Hom(V, V^{∗}) = 𝕂 ⊕ ker(T r_{g}) is an orthogonal direct sum, with operators P′′_{1} = α · T r_{g}, Q′′_{1} : λ ↦ λ · β · g, with α · β · T r_{V} (Id_{V}) = 1 as in Example 2.9. Also, P′′_{2} = Id_{Hom(V,V^{∗})} − Q′′_{1} ∘ P′′_{1}, and Q′′_{2} is just the inclusion of the subspace ker(T r_{g}) in Hom(V, V^{∗}). By Example 1.77, Hom(V, V^{∗}) ⊗ W is a direct sum of 𝕂 ⊗ W and (ker(T r_{g})) ⊗ W, with operators P_{i} = [P′′_{i} ⊗ Id_{W}], Q_{i} = [Q′′_{i} ⊗ Id_{W}]. Let H = Hom(Id_{V} , k_{V W}) ∘ n_{1}, and let P′_{1} = α · T r_{g;W}, so that the following diagram is commutative by Lemma 4.26.

\xymatrix{Hom(V, Hom(V, W)) \ar[d]^{α·T r_{g;W}} & Hom(V, V^{∗}) ⊗ W \ar[l]^{H} \ar[d]^{[(α·T r_{g})⊗Id_{W}]} \\ W & 𝕂 ⊗ W \ar[l]^{l_{W}}}

Let Q′_{2} be the inclusion of ker(T r_{g;W}) in Hom(V, Hom(V, W)), which is a linear monomorphism so that P′_{1}∘Q′_{2} = 0_{Hom(ker(T r_{g;W}),W)}. Define H_{2} : (ker(T r_{g}))⊗W → ker(T r_{g;W}) by H_{2} = H ∘Q_{2}; the image of H_{2} is contained in ker(T r_{g;W}) by Lemma 4.26, so Q′_{2} ∘ H_{2} = H ∘ Q_{2}. Theorem 1.98 applies, so that Hom(V, Hom(V, W)) has a direct sum structure W ⊕ ker(T r_{g;W}), and H respects the direct sums. By Theorem 3.60, if W has a metric y, then the direct sum Hom(V, V^{∗}) ⊗ W = (𝕂 ⊗ W) ⊕ ((ker(T r_{g})) ⊗ W) is orthogonal with respect to the induced metric. Since H is an isometry (as mentioned in the proof of the previous Theorem), it follows from Theorem 3.61 that W ⊕ ker(T r_{g;W}) is orthogonal with respect to the induced metric.

Corollary 4.41. The metric induced on W by the direct sum from the previous Theorem is β^{2} · T r_{V} (Id_{V}) · y.

Proof. The induced metric on 𝕂 ⊗ W is {h^{ν} ⊗ y}, for ν = β^{2} · T r_{V} (Id_{V}) by Theorem 3.60 and Lemma 3.71. By Theorem 3.61, P′_{1} ∘ H ∘ Q_{1} = l_{W} ∘ P_{1} ∘ Q_{1} = l_{W} : 𝕂 ⊗ W → W is an isometry, so by Lemma 3.69, the metric in the target must be ν · y.

Corollary 4.42. Given a metric g on V, if both \frac{1}{2} ∈ 𝕂 and T r_{V} (Id_{V}) ≠ 0, then there is a direct sum Hom(V, Hom(V, W)) = W ⊕ Sym_{0}(g; W) ⊕ Alt(V; W), where Sym_{0}(g; W) is the kernel of the restriction of T r_{g;W} to Sym(V; W). If W has a metric y, then there is an orthogonal direct sum.

Exercise 4.43. For K : V → Hom(V, W), an orthogonal direct sum V = V_{1} ⊕ V_{2} with respect to a metric g on V, and the induced metrics g_{1}, g_{2}, on V_{1}, V_{2},

T r_{g;W} (K) = T r_{g_{1};W} (Hom(Q_{1}, Id_{W}) ∘ K ∘ Q_{1}) + T r_{g_{2};W} (Hom(Q_{2}, Id_{W}) ∘ K ∘ Q_{2}).

Hint. In analogy with Exercise 3.101, Lemma 1.58 and Corollary 2.57 apply:

LHS = T r_{V^{∗};W} (k_{V W}^{−1} ∘ K ∘ g^{−1})
= T r_{V^{∗};W} (k_{V W}^{−1} ∘ K∘(Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2}) ∘ g^{−1}∘(Q_{1} ∘ P_{1} + Q_{2} ∘ P_{2})^{∗})
= T r_{V^{∗};W} (k_{V W}^{−1} ∘ K ∘ Q_{1} ∘ P_{1} ∘ g^{−1} ∘ P_{1}^{∗} ∘ Q_{1}^{∗})
+T r_{V^{∗};W} (k_{V W}^{−1} ∘ K ∘ Q_{2} ∘ P_{2} ∘ g^{−1} ∘ P_{2}^{∗} ∘ Q_{2}^{∗})
= T r_{V_{1};W} ([Q_{1}^{∗} ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ K ∘ Q_{1} ∘ g_{1}^{−1})
+T r_{V_{2};W} ([Q_{2}^{∗} ⊗ Id_{W}] ∘ k_{V W}^{−1} ∘ K ∘ Q_{2} ∘ g_{2}^{−1})
= T r_{V_{1};W} (k_{V_{1}W}^{−1} ∘ Hom(Q_{1}, Id_{W}) ∘ K ∘ Q_{1} ∘ g_{1}^{−1})
+T r_{V_{2};W} (k_{V_{2}W}^{−1} ∘ Hom(Q_{2}, Id_{W}) ∘ K ∘ Q_{2} ∘ g_{2}^{−1}) = RHS.

4.4. Revisiting the generalized trace

We return to some notions introduced in Section 2.4. Recall, from Notation 2.67, the map η_{V} = s ∘ k^{−1} ∘ Q_{1}^{1} : 𝕂 → V ⊗ V^{∗}.

Notation 4.44. For finite-dimensional V, consider the following diagram.

[figure: End(V) ⊗ U; n_{1}; V^{∗} ⊗ V ⊗ U; [k⊗Id_{U}]; [s⊗Id_{U}]; k_{V,V ⊗U}; 𝕂 ⊗ U; [Q_{1}^{1}⊗Id_{U}]; [η_{V} ⊗Id_{U}]; l_{U}; V ⊗ V^{∗} ⊗ U; [Id_{V} ⊗k_{V U}]; U; η_{V U}; V ⊗ Hom(V, U); n_{2}; Hom(V, V ⊗ U)]

The top block is from (2.9), the back triangle is commutative by Lemma 1.61, and the right block is also commutative by a variation on Lemma 1.61. So, a map η_{V U} : U → V ⊗ Hom(V, U) can be defined by the following equal formulas:

η_{V U} = [Id_{V} ⊗ k_{V U}] ∘ [η_{V} ⊗ Id_{U}] ∘ l_{U}^{−1}
= n_{2}^{−1} ∘ n_{1} ∘ [Q_{1}^{1} ⊗ Id_{U}] ∘ l_{U}^{−1} :
u : ↦ n_{2}^{−1}(n_{1}(Id_{V} ⊗ u)).

With the above notation, Theorem 2.75 can be re-stated in terms of η_{V U}.

Corollary 4.45. For finite-dimensional V, n_{2} as in the above diagram, any F : V ⊗ U → V ⊗ W, and u ∈ U,

(T r_{V;U,W} (F))(u) = T r_{V;W} (F ∘ (n_{2}(η_{V U}(u)))).

Proof. The following diagram is a modification of the diagram from the Proof of Theorem 2.75.

\xymatrix{End(V) ⊗ U & V^{∗} ⊗ V ⊗ U \ar[l]^{[k⊗Id_{U}]} \ar[r]^{[Id_{V ^{∗}} ⊗F]} \ar[d]_{[s⊗Id_{U}]} \ar[ddr]^{k_{V,V ⊗U}} & V^{∗} ⊗ V ⊗ W \ar[d]^{k_{V,V ⊗W}} \ar[r]^{[Ev_{V} ⊗Id_{W}]} & 𝕂 ⊗ W \ar[d]^{l_{W}} \\ 𝕂 ⊗ U \ar[u]_{[Q_{1}^{1}⊗Id_{U}]} \ar[d]^{l_{U}} \ar[r]^{[η_{V} ⊗Id_{U}]} & V ⊗ V^{∗} ⊗ U \ar[d]_{[Id_{V} ⊗k_{V U}]} & Hom(V, V ⊗ W) \ar[r]^{T r_{V ;W}} & W \\ U \ar[r]^{η_{V U}} & V ⊗ Hom(V, U) \ar[r]^{n_{2}} & Hom(V, V ⊗ U) \ar[u]_{Hom(Id_{V} ,F)} & Hom(V, W) ⊗ V \ar[ul]_{n′} \ar[u]_{Ev_{V W}}}

The diagram is commutative; the left blocks and lower middle triangle by the construction of η_{V} and η_{V U} in Notation 4.44, the upper middle triangle by Lemma 1.58, and the right block copied from the Proof of Theorem 2.75. The path from U to W along the top row is T r_{V;U,W} (F) by Theorem 2.68, and equals the same composite map from U to W along the lower row, so

(4.9)  T r_{V;U,W} (F) : u ↦ Ev_{V W} ((n′)^{−1}(F ∘ (n_{2}(η_{V U}(u)))))
= T r_{V;W} (F ∘ (n_{2}(η_{V U}(u))))
= T r_{V;W} (F ∘ (n_{1}(Id_{V} ⊗ u))).

The composition in (4.9) from the lower path in the diagram, or equivalently

(4.10)  T r_{V;U,W} (F) = Ev_{V W} ∘ (n′)^{−1} ∘ Hom(Id_{V} , F) ∘ n_{2} ∘ η_{V U} : U → W,

has two interesting properties: it does not involve scalar multiplication or duals (except in the construction of η_{V U}), and the maps η_{V U} and Ev_{V W} appear in symmetric roles.

Exercise 4.46. In the case U = 𝕂, these maps are equal: η_{V U} = η_{V 𝕂} = η_{V}.

The following pair of Theorems are analogues of Theorem 2.95; the idea is that η_{V U} and Ev_{V W} satisfy identities analogous to the abstractly defined evaluation and coevaluation maps as in Definition 2.96. Theorem 4.48 uses the transpose for vector valued forms from Definition 4.2.

Theorem 4.47. For any U and V, if V is finite-dimensional, and η_{V U} : U → V ⊗Hom(V, U) is defined as in Notation 4.44, then the following composite is equal to a switching map:

[Id_{V} ⊗ Ev_{V U}] ∘ [η_{V U} ⊗ Id_{V}] = s_{0} : U ⊗ V → V ⊗ U.

Proof. The claim is analogous to the first identity from Theorem 2.95, and the proof is also analogous. The labeling V = V_{1} = V_{2} = V_{3} is introduced to track the action of the switching s maps. The upper and middle left squares are from the diagram from Notation 4.44. The claim is that the lower left triangle in the following diagram is commutative.

\xymatrix{Hom(V_{2}, V_{1}) ⊗ U ⊗ V_{3} & V_{2}^{∗} ⊗ V_{1} ⊗ U ⊗ V_{3} \ar[r]_{[Id_{V ^{∗}} ⊗s_{4}^{−1}]} \ar[d]^{[[s⊗Id_{U}]⊗Id_{V} ]} \ar[l]^{[[k⊗Id_{U}]⊗Id_{V} ]} & V_{2}^{∗} ⊗ V_{3} ⊗ V_{1} ⊗ U \ar[dd]_{[Ev_{V} ⊗Id_{V ⊗U}]} \\ 𝕂 ⊗ U ⊗ V_{3} \ar[u]_{[[Q_{1}^{1}⊗Id_{U}]⊗Id_{V} ]} \ar[r]^{[[η_{V} ⊗Id_{U}]⊗Id_{V} ]} \ar[d]^{[l_{U} ⊗Id_{V} ]} & V_{1} ⊗ V_{2}^{∗} ⊗ U ⊗ V_{3} \ar[d]^{[[Id_{V} ⊗k_{V U}]⊗Id_{V} ]} \\ U ⊗ V_{3} \ar[r]^{[η_{V U} ⊗Id_{V} ]} \ar[dr]^{s_{0}} & V_{1} ⊗ Hom(V_{2}, U) ⊗ V_{3} \ar[d]^{[Id_{V} ⊗Ev_{V U}]} & 𝕂 ⊗ V_{1} ⊗ U \ar[dl]^{l_{V ⊗U}} \\  & V ⊗ U}

The commutativity of the right block is easy to check, where the switching map s_{4} is as in Theorem 2.86 and Corollary 2.87. The composition starting at U ⊗ V in the lower left and going all the way around the diagram clockwise to V ⊗ U is the trace, as in Theorem 2.68, of s_{4}^{−1}, and the computation of Example 2.91 applies.

[Id_{V} ⊗ Ev_{V U}] ∘ [η_{V U} ⊗ Id_{V}]
= l_{V ⊗U} ∘ [Ev_{V} ⊗ Id_{V ⊗U}] ∘ [Id_{V^{∗}} ⊗ s_{4}^{−1}] ∘ [(k^{−1} ∘ Q_{1}^{1}) ⊗ Id_{U⊗V}] ∘ [l_{U}^{−1} ⊗ Id_{V}]
= T r_{V;U⊗V,V ⊗U}(s_{4}^{−1}) = s_{3}^{−1} = s_{0}.

Theorem 4.48. For any U and V, if V is finite-dimensional, then the n maps indicated in the following diagram are invertible,

\xymatrix{Hom(V, U) \ar[r]^{Hom(Id_{V} ,η_{V U})} & Hom(V, V ⊗ Hom(V, U)) & Hom(V, Hom(V, U)) ⊗ V \ar[l]^{n_{1}} \ar[d]_{[T_{V ;U} ⊗Id_{V} ]} \\  & Hom(V, Hom(V, U) ⊗ V) \ar[ul]^{Hom(Id_{V} ,Ev_{V U})} & Hom(V, Hom(V, U)) ⊗ V \ar[l]^{n_{2}}}

and the diagram is commutative in the sense that this composite map is equal to the identity map:

Hom(Id_{V} , Ev_{V U}) ∘ n_{2} ∘ [T_{V;U} ⊗ Id_{V}] ∘ n_{1}^{−1} ∘ Hom(Id_{V} , η_{V U}) = Id_{Hom(V,U)}.

Proof. The claim is analogous to the second identity from Theorem 2.95, and the overall proof is also analogous. As in the Proof of Theorem 4.47, the labeling V = V_{1} = V_{2} = V_{3} is introduced to track the action of the n, p, and s maps in this “main diagram.”

\xymatrix{ &  & V_{1}^{∗} ⊗ U \ar[ddll]^{k_{V U}} \\  &  & 𝕂 ⊗ V_{1}^{∗} ⊗ U \ar[u]^{l_{V ^{∗}⊗U}} \ar[d]_{[Q_{1}^{1}⊗Id_{V ^{∗}⊗U}]} \\ Hom(V_{1}, U) \ar[d]^{Hom(Id_{V} ,η_{V U})} & V_{2}^{∗} ⊗ V_{3} ⊗ V_{1}^{∗} ⊗ U \ar[r]_{[k⊗Id_{V ^{∗}⊗U}]} \ar[ddr]^{[p⊗Id_{V ^{∗}⊗U}]} & Hom(V_{2},V_{3})⊗ V_{1}^{∗}⊗U \ar[d]^{[t⊗Id_{V ^{∗}⊗U}]} \\ Hom(V_{1}, V_{3}⊗ Hom(V_{2}, U)) & V_{1}^{∗} ⊗ V_{3} ⊗ V_{2}^{∗} ⊗ U \ar[l]^{a_{1}} \ar[u]^{[s′′⊗Id_{U}]} \ar[d]_{s_{1}} & Hom(V_{3}^{∗},V_{2}^{∗})⊗ V_{1}^{∗}⊗U \\ Hom(V_{1}, Hom(V_{2}, U))⊗ V_{3} \ar[u]^{n_{1}} \ar[d]^{[T_{V ;U} ⊗Id_{V} ]} & V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ U ⊗ V_{3} \ar[ddl]^{a_{2}} \ar[r]_{p′} & V_{3}^{∗∗} ⊗ V_{2}^{∗} ⊗ V_{1}^{∗} ⊗ U \ar[d]_{[Id_{V ^{∗∗}} ⊗s_{4}]} \ar[u]_{[k′⊗Id_{V ^{∗}⊗U}]} \\ Hom(V_{2}, Hom(V_{1}, U))⊗ V_{3} \ar[d]^{n_{2}} &  & V_{3}^{∗∗} ⊗ V_{1}^{∗} ⊗ U ⊗ V_{2}^{∗} \ar[d]_{[Ev_{V ^{∗}} ⊗Id_{U⊗V ^{∗}}]} \\ Hom(V_{2}, Hom(V_{1}, U)⊗ V_{3}) \ar[d]^{Hom(Id_{V} ,Ev_{V U})} &  & 𝕂 ⊗ U ⊗ V_{2}^{∗} \ar[d]_{l_{U⊗V ^{∗}}} \\ Hom(V_{2}, U) & V_{2}^{∗} ⊗ U \ar[l]^{k_{V U}} & U ⊗ V_{2}^{∗} \ar[l]^{s_{3}^{−1}}}

The abbreviations t_{V V} = t, k_{V V} = k, k_{V^{∗}V^{∗}} = k′, and the map p are as in the notation from Lemma 2.5. The switching involution s′′ appeared in Lemma 2.82. The composite t ∘ Q_{1}^{1} : 𝕂 → End(V^{∗}) is equal to another inclusion, temporarily denoted Q̃_{1}^{1}, that maps 1 to Id_{V^{∗}} = Id_{V}^{∗} = t(Id_{V}) as in Equation (2.4). So, the composition in the right column is the trace, as in Theorem 2.68, of s_{4}, and the computation of Example 2.90 applies:

l_{U⊗V^{∗}} ∘ [Ev_{V^{∗}} ⊗ Id_{U⊗V^{∗}}] ∘ [Id_{V^{∗∗}} ⊗ s_{4}] ∘ [((k′)^{−1} ∘ Q̃_{1}^{1}) ⊗ Id_{V^{∗}⊗U}] ∘ l_{V^{∗}⊗U}^{−1}

= T r_{V;V^{∗}⊗U,U⊗V^{∗}}(s_{4}) = s_{3}.

The p, k, and n maps are invertible by the finite-dimensionality of V; in the right center block, the triangle with k, t, and p is commutative by Lemma 1.71, and it is easy to check that the other triangle with p′ is also commutative. The claim of the Theorem is that the composition in the left column gives the identity map; this will follow if we can find a_{1} and a_{2} as indicated that make the main diagram commutative.

The following maps, and s_{1} in the above diagram, are from Lemma 4.10, in the case V = V_{1} = V_{2} = V_{3}, U = W:

a_{1} = Hom(Id_{V} , [Id_{V} ⊗ k_{V U}]) ∘ k_{V,V ⊗V^{∗}⊗U}
a_{2} = Hom(Id_{V} , [k_{V U} ⊗ Id_{V}]) ∘ k_{V,V^{∗}⊗U⊗V} .

The commutativity of the left center block then follows from Lemma 4.10, so to prove the Theorem it only remains to show that these maps a_{1}, a_{2} make the upper and lower blocks of the main diagram commutative.

To check the lower block, start with ψ ⊗ ϕ ⊗ u ⊗ v ∈ V^{∗} ⊗ V^{∗} ⊗ U ⊗ V at its top, and w ∈ V.

Hom(Id_{V} , Ev_{V U}) ∘ Hom(Id_{V} , [k_{V U} ⊗ Id_{V}]) ∘ k_{V,V^{∗}⊗U⊗V} :
ψ ⊗ ϕ ⊗ u ⊗ v ↦ Ev_{V U} ∘ [k_{V U} ⊗ Id_{V}] ∘ (k_{V,V^{∗}⊗U⊗V} (ψ ⊗ ϕ ⊗ u ⊗ v)) :
w ↦ Ev_{V U}([k_{V U} ⊗ Id_{V}](ψ(w) · ϕ ⊗ u ⊗ v))
= Ev_{V U}(ψ(w) · (k_{V U}(ϕ ⊗ u)) ⊗ v) = ψ(w) · ϕ(v) · u,
k_{V U} ∘ s_{3}^{−1} ∘ l_{U⊗V^{∗}} ∘ [Ev_{V^{∗}} ⊗ Id_{U⊗V^{∗}}] ∘ [Id_{V^{∗∗}} ⊗ s_{4}] ∘ p′ :
ψ ⊗ ϕ ⊗ u ⊗ v ↦ (k_{V U} ∘ s_{3}^{−1} ∘ l_{U⊗V^{∗}} ∘ [Ev_{V^{∗}} ⊗ Id_{U⊗V^{∗}}])((d_{V} (v)) ⊗ ϕ ⊗ u ⊗ ψ)
= (k_{V U} ∘ s_{3}^{−1})(ϕ(v) · u ⊗ ψ) = ϕ(v) · k_{V U}(ψ ⊗ u) :
w ↦ ϕ(v) · ψ(w) · u.

In the following diagram, the two lower right commutative squares are from the definition of η_{V U}.

\xymatrix{ & V_{2}^{∗} ⊗ V_{3} ⊗ V_{1}^{∗} ⊗ U \ar[d]^{[k⊗Id_{V ^{∗}⊗U}]} \\  & Hom(V_{2}, V_{3}) ⊗ V_{1}^{∗} ⊗ U \ar[d]^{[s_{5}⊗Id_{U}]} & V_{1}^{∗} ⊗ V_{3} ⊗ V_{2}^{∗} ⊗ U \ar[ul]_{[s′′⊗Id_{U}]} \ar[d]_{k_{V,V ⊗V ^{∗}⊗U}} \\ 𝕂 ⊗ V_{1}^{∗} ⊗ U \ar[dd]^{l_{V ^{∗}⊗U}} \ar[ur]^{[Q_{1}^{1}⊗Id_{V ^{∗}⊗U}]} & V_{1}^{∗} ⊗ Hom(V_{2}, V_{3}) ⊗ U \ar[d]^{k_{V,End(V )⊗U}} & Hom(V, V_{3} ⊗ V_{2}^{∗} ⊗ U) \\  & Hom(V_{1}, Hom(V_{2}, V_{3}) ⊗ U) & Hom(V_{1}, V_{2}^{∗} ⊗ V_{3} ⊗ U) \ar[l]^{Hom(Id_{V} ,[k⊗Id_{U}])} \ar[d]_{Hom(Id_{V} ,[s⊗Id_{U}])} \ar[u]^{Hom(Id_{V} ,[s⊗Id_{U}])} \\ V_{1}^{∗} ⊗ U \ar[dr]_{k_{V U}} & Hom(V_{1}, 𝕂 ⊗ U) \ar[r]_{Hom(Id_{V} ,[η_{V} ⊗Id_{U}])} \ar[d]^{Hom(Id_{V} ,l_{U})} \ar[u]^{Hom(Id_{V} ,[Q_{1}^{1}⊗Id_{U}])} & Hom(V_{1}, V_{3} ⊗ V_{2}^{∗} ⊗ U) \ar[d]_{Hom(Id_{V} ,[Id_{V} ⊗k_{V U}])} \\  & Hom(V_{1}, U) \ar[r]_{Hom(Id_{V} ,η_{V U})} & Hom(V_{1}, V_{3} ⊗ Hom(V_{2}, U))}

To check the commutativity of the upper right block, start with ϕ ⊗ v ⊗ ψ ⊗ u ∈ V_{1}^{∗} ⊗ V_{3} ⊗ V_{2}^{∗} ⊗ U and w ∈ V:

Hom(Id_{V} , [k ⊗ Id_{U}]) ∘ Hom(Id_{V} , [s^{−1} ⊗ Id_{U}]) ∘ k_{V,V ⊗V^{∗}⊗U} :

ϕ ⊗ v ⊗ ψ ⊗ u ↦ [(k ∘ s^{−1}) ⊗ Id_{U}] ∘ (k_{V,V ⊗V^{∗}⊗U}(ϕ ⊗ v ⊗ ψ ⊗ u)) :
w ↦ [(k ∘ s^{−1}) ⊗ Id_{U}](ϕ(w) · v ⊗ ψ ⊗ u) = ϕ(w) · (k(ψ ⊗ v)) ⊗ u,
k_{V,End(V)⊗U} ∘ [s_{5} ⊗ Id_{U}] ∘ [k ⊗ Id_{V^{∗}⊗U}] ∘ [s′′ ⊗ Id_{U}] :
ϕ ⊗ v ⊗ ψ ⊗ u ↦ k_{V,End(V)⊗U}(ϕ ⊗ (k(ψ ⊗ v)) ⊗ u) :
w ↦ ϕ(w) · (k(ψ ⊗ v)) ⊗ u.

To check the commutativity of the left block, start with α ⊗ ϕ ⊗ u ∈ 𝕂 ⊗ V^{∗} ⊗ U:

k_{V,End(V)⊗U} ∘ [s_{5} ⊗ Id_{U}] ∘ [Q_{1}^{1} ⊗ Id_{V^{∗}⊗U}] :
α ⊗ ϕ ⊗ u ↦ k_{V,End(V)⊗U}(ϕ ⊗ (α · Id_{V}) ⊗ u) :
v ↦ (ϕ(v)) · (α · Id_{V}) ⊗ u,
Hom(Id_{V} , [Q_{1}^{1} ⊗ Id_{U}]) ∘ Hom(Id_{V} , l_{U}^{−1}) ∘ k_{V U} ∘ l_{V^{∗}⊗U} :
α ⊗ ϕ ⊗ u ↦ [Q_{1}^{1} ⊗ Id_{U}] ∘ l_{U}^{−1} ∘ (k_{V U}(α · ϕ ⊗ u)) :
v ↦ (α · ϕ(v)) · Id_{V} ⊗ u.

The downward composite of the four arrows in the right column equals the previously defined a_{1}. So, the above calculation is enough to establish the commutativity of the top block in the main diagram:

Hom(Id_{V} , η_{V U}) ∘ k_{V U} = a_{1} ∘ [(s′′)^{−1} ⊗ Id_{U}] ∘ [(k^{−1} ∘ Q_{1}^{1}) ⊗ Id_{V^{∗}⊗U}] ∘ l_{V^{∗}⊗U}^{−1}.

As mentioned earlier, this proves the claim of the Theorem.

4.5. Topics and applications

4.5.1. Quadratic forms.

Proposition 4.49. Given vector spaces V, W, and a function 𝔮 : V ⇝ W, if \frac{1}{2} ∈ 𝕂 then the following are equivalent.

(1) There exists a symmetric W -valued form h_{1} : V → Hom(V, W) such that

for all v ∈ V,

𝔮(v) = (h_{1}(v))(v).

(2) There exists a W -valued bilinear form h_{2} : V → Hom(V, W) such that for

all v ∈ V,

𝔮(v) = (h_{2}(v))(v).

(3) There exists a bilinear function B_{1} : V × V ⇝ W such that for all v ∈ V,

𝔮(v) = B_{1}(v, v).

(4) For any α ∈ 𝕂 and v ∈ V, 𝔮(α · v) = α^{2} · 𝔮(v), and the function B_{2} : V × V ⇝ W defined by

B_{2}(u, v) = 𝔮(u + v) − 𝔮(u) − 𝔮(v)

is bilinear.

(5) For any α ∈ 𝕂 and v ∈ V, 𝔮(α · v) = α^{2} · 𝔮(v), and the function B_{3} : V × V ⇝ W defined by

B_{3}(u, v) = 𝔮(u) + 𝔮(v) − 𝔮(u − v)

is bilinear.

(6) For any α ∈ 𝕂 and v ∈ V, 𝔮(α · v) = α^{2} · 𝔮(v), and the function B_{4} : V × V ⇝ W defined by

B_{4}(u, v) = 𝔮(u + v) − 𝔮(u − v)

is bilinear.

(7) For all u, v ∈ V, 𝔮 satisfies:

(4.11)  𝔮(u + v) + 𝔮(u − v) = 2 · 𝔮(u) + 2 · 𝔮(v),

and the function B_{4} : V × V ⇝ W defined by

B_{4}(u, v) = 𝔮(u + v) − 𝔮(u − v)

satsifies, for all α ∈ 𝕂, B_{4}(α · u, v) = α · B_{4}(u, v).

Proof. As in Notation 0.36, the ⇝ arrow symbol refers to functions which are not necessarily linear. The functions B_{1}, …, B_{4} are bilinear as in Definition 1.23.

The implication (1) ⟹ (2) is trivial. For (2) ⟹ (3), define B_{1} by B_{1}(u, v) = (h_{2}(u))(v), and similarly for (3) ⟹ (2), define h_{2} by the formula h_{2}(u) : v ↦ B_{1}(u, v). The correspondence between h_{2} and B_{1} is a W -valued version of the construction from Example 1.51.

Now, assuming (3), so that B_{1} is bilinear and 𝔮(v) = B_{1}(v, v), the first property from (4), (5) and (6) is immediate:

𝔮(α · v) = B_{1}(α · v, α · v) = α^{2} · B_{1}(v, v) = α^{2} · 𝔮(v).

Expanding B_{2}, B_{3}, B_{4} in terms of B_{1}:

B_{2}(u, v) = 𝔮(u + v) − 𝔮(u) − 𝔮(v)
= B_{1}(u + v, u + v) − B_{1}(u, u) − B_{1}(v, v)
= B_{1}(u, v) + B_{1}(v, u).
B_{3}(u, v) = 𝔮(u) + 𝔮(v) − 𝔮(u − v)
= B_{1}(u, u) + B_{1}(v, v) − B_{1}(u − v, u − v)
= B_{1}(u, v) + B_{1}(v, u).
B_{4}(u, v) = 𝔮(u + v) − 𝔮(u − v)
= B_{1}(u + v, u + v) − B_{1}(u − v, u − v)
(4.12)  = B_{1}(u, v) + B_{1}(v, u) + B_{1}(u, v) + B_{1}(v, u).

The bilinearity of B_{1} implies the bilinearity of B_{1}(u, v) + B_{1}(v, u), so (3) implies (4), (5), and (6). The relation B_{4} = B_{2} + B_{3} also shows that any two of (4), (5), and (6) together imply the third.

For (3) ⟹ (7), expanding (4.11) in terms of B_{1} shows LHS = RHS (a related quantity already appeared in Proposition 3.125), and B_{4} is bilinear as in (4.12).

So far, the implications have not yet used \frac{1}{2} ∈ 𝕂. To show (6) ⟹ (3), given the bilinear form B_{4}, define B_{1}(u, v) = (\frac{1}{2})^{2} · B_{4}(u, v), so that B_{1} is bilinear, and for any v ∈ V,

B_{1}(v, v) = (\frac{1}{2})^{2} · (𝔮(v + v) − 𝔮(v − v))

= (\frac{1}{2})^{2} · (𝔮((1 + 1) · v) − 𝔮(0 · v))

(4.13)  = (\frac{1}{2})^{2} · ((1 + 1)^{2} · 𝔮(v) − 0^{2} · 𝔮(v)) = 𝔮(v).

Similar calculations using \frac{1}{2} ∈ 𝕂 would directly show (4) ⟹ (3) and (5) ⟹ (3).

For (7) ⟹ (3), Equation (4.11) with u = v = 0_{V} gives 𝔮(0_{V}) = 0_{W}, and with u = 0_{V} gives 𝔮(−v) = 𝔮(v). Then

B_{4}(v, u) = 𝔮(v + u) − 𝔮(v − u) = 𝔮(v + u) − 𝔮(u − v) = B_{4}(u, v).

It follows that B_{4}(u, α · v) = B_{4}(α · v, u) = α · B_{4}(v, u) = α · B_{4}(u, v). The following calculation shows that 2·B_{4} is additive in the first entry, using (4.11) in steps (4.15) and (4.17) and some add-and-subtract steps in (4.14) and (4.16).

2 · B_{4}(u_{1} + u_{2}, v)
= 2 · 𝔮(u_{1} + u_{2} + v) − 2 · 𝔮(u_{1} + u_{2} − v)
= 2 · 𝔮(u_{1} + u_{2} + v) + 2 · 𝔮(u_{1} − v) − 2 · 𝔮(u_{1} + u_{2} − v) − 2 · 𝔮(u_{1} + v)

+2 · 𝔮(u_{1} + v) − 2 · 𝔮(u_{1} − v) (4.14)

= 𝔮(2 · u_{1} + u_{2}) + 𝔮(u_{2} + 2 · v) − 𝔮(2 · u_{1} + u_{2}) − 𝔮(u_{2} − 2 · v)

+2 · 𝔮(u_{1} + v) − 2 · 𝔮(u_{1} − v) (4.15)

= 𝔮(u_{2} + 2 · v) + 𝔮(u_{2}) − 𝔮(u_{2} − 2 · v) − 𝔮(u_{2})

+2 · 𝔮(u_{1} + v) − 2 · 𝔮(u_{1} − v) (4.16)

= 2 · 𝔮(u_{2} + v) + 2 · 𝔮(v) − 2 · 𝔮(u_{2} − v) − 2 · 𝔮(−v)

+2 · 𝔮(u_{1} + v) − 2 · 𝔮(u_{1} − v) (4.17)

= 2 · B_{4}(u_{1}, v) + 2 · B_{4}(u_{2}, v).

By symmetry again, 2 · B_{4} is bilinear, and so is B_{1} = (\frac{1}{2})^{2} · B_{4}. The following calculation establishing (3) using (4.11) is different from (4.13):

B_{1}(v, v) = (\frac{1}{2})^{2} · (𝔮(v + v) − 𝔮(v − v))

= (\frac{1}{2})^{2} · (𝔮(v + v) + 𝔮(v − v))

= (\frac{1}{2})^{2} · (2 · 𝔮(v) + 2 · 𝔮(v)) = 𝔮(v).

Finally, to show (2) ⟹ (1) using \frac{1}{2} ∈ 𝕂, let h_{1} be the symmetric part of h_{2} as in (3.2) and (4.6):

(h_{1}(u))(v) = \frac{1}{2} · ((h_{2}(u))(v) + (h_{2}(v))(u))
⟹ (h_{1}(v))(v) = \frac{1}{2} · (𝔮(v) + 𝔮(v)) = 𝔮(v).

Definition 4.50. Assuming \frac{1}{2} ∈ 𝕂, a function 𝔮 : V ⇝ W satisfying any of the equivalent properties from Proposition 4.49 is a W -valued \underline{quadratic form}.

Remark 4.51. The equations from (4), (5), and (6) are known as \underline{polarization} formulas. Equation (4.11) from (7) is the \underline{parallelogram law} for 𝔮. The case where \frac{1}{2} ∉ 𝕂 is more complicated and not considered here; the remaining statements here in Section 4.5.1 will all assume \frac{1}{2} ∈ 𝕂.

Exercise 4.52. Given a quadratic form 𝔮, the symmetric form h_{1} from Proposition 4.49 is unique.

Hint. This is a statement about symmetric forms rather than quadratic forms: the claim is that if h_{0} and h_{1} are both symmetric forms and (h_{0}(v))(v) = (h_{1}(v))(v) for all v ∈ V, then h_{0} = h_{1}. The hint is to expand (h_{0}(u + v))(u + v) and use \frac{1}{2} ∈ 𝕂.

Exercise 4.53. The set 𝔔(V; W) of W -valued quadratic forms on V is a vector space. The map 𝔣_{V W} defined by 𝔣_{V W} : 𝔔(V; W) → Sym(V; W) : 𝔮 ↦ h_{1} as in Exercise 4.52 is linear and invertible.

Hint. 𝔔(V; W) is a subspace of F(V, W) as in Example 6.28.

Definition 4.54. For vector spaces U, V, W, define a linear map 𝔱_{UV}^{W} by:

𝔱_{UV}^{W} : Hom(U, V) → Hom(𝔔(V; W), 𝔔(U; W))
H ↦ (𝔮 ↦ 𝔮 ∘ H).

Exercise 4.55. There are a few things to check in Definition 4.54: first, that for H : U → V, the composite 𝔮 ∘ H is a quadratic form, second, that 𝔱_{UV}^{W} (H) is linear, and third, that 𝔱_{UV}^{W} is linear. Further, the following diagram is commutative.

\xymatrix{𝔔(V; W) \ar[r]^{𝔱_{UV}^{W} (H)} \ar[d]^{𝔣_{V W}} & 𝔔(U; W) \ar[d]^{𝔣_{UW}} \\ Sym(V; W) \ar[r] \ar@{^{(}->}[d] & Sym(U; W) \ar@{^{(}->}[d] \\ Hom(V, Hom(V, W)) \ar[r]^{Hom(H,Hom(H,Id_{W}))} & Hom(U, Hom(U, W))}

The middle horizontal arrow is the map induced by Hom(H, Hom(H, Id_{W})) as in Lemma 4.17.

Hint. The linearity of 𝔱_{UV}^{W} (H) can be checked directly, but also follows from the commutativity of the upper block in the diagram. It is enough to check the commutativity of the large block from upper left to lower right; temporarily denote the left inclusion Q_{V} and the right inclusion Q_{U}. For 𝔮 with 𝔣_{V W} (𝔮) = h_{1},

Hom(H, Id_{W}) ∘ (Q_{V} (𝔣_{V W} (𝔮))) ∘ H : v ↦ (h_{1}(H(v))) ∘ H :
(4.18)  u ↦ (h_{1}(H(v)))(H(u)).

Then Q_{U} ∘ 𝔣_{UW} ∘ (𝔱_{UV}^{W} (H)) : 𝔮 ↦ Q_{U}(𝔣_{UW} (𝔮 ∘ H)) = h_{0} is the unique W -valued bilinear form that is symmetric and that has the property (𝔮 ∘ H)(u) = (h_{0}(u))(u). However, 𝔮(H(u)) = (h_{1}(H(u)))(H(u)), so Hom(H, Hom(H, Id_{W}))(Q_{V} (h_{1})) from (4.18) has both these properties and is equal to h_{0} by uniqueness, proving the commutativity of the diagram.

Exercise 4.56. 𝔱_{V V}^{W} (Id_{V}) = Id_{𝔔 (V;W)}. For H : U → V and A : V → X,
𝔱_{UX}^{W}(A ∘ H) = 𝔱_{UV}^{W} (H) ∘ 𝔱_{V X}^{W}(A).

In particular, if H has a linear left (or right) inverse, then 𝔱_{UV}^{W} (H) has a linear right (or left) inverse.

Proposition 4.57. ([HK] §10.2) Suppose U is finite-dimensional and W ≠ {0_{W} }. For H : U → U, the following are equivalent.

(1) 𝔱_{UU}^{W}(H) : 𝔔(U; W) → 𝔔(U; W) is one-to-one.

(2) H is invertible.

Proof. The (2) ⟹ (1) direction follows from Exercise 4.56.

For the other direction, suppose, contrapositively, that H is not invertible; then by Lemma 1.19 (which uses the finite-dimensional property of U), H^{∗} : U^{∗} → U^{∗} is not invertible. By Claim 0.50 (using the finite-dimensional property of U^{∗}), H^{∗} is not one-to-one and there exists ϕ ∈ U^{∗} so that ϕ ≠ 0_{U^{∗}} and H^{∗}(ϕ) = ϕ ∘ H = 0_{U^{∗}}. Pick any w ∈ W with w ≠ 0_{W}, and define a bilinear form

g = k_{U,Hom(U,W)}(ϕ ⊗ (k_{UW} (ϕ ⊗ w))) ∈ Hom(U, Hom(U, W)).

g ≠ 0_{Hom(U,Hom(U,W))} because there is some x ∈ U with ϕ(x) ≠ 0, and

(g(x))(x) = (ϕ(x) · k_{UW} (ϕ ⊗ w))(x) = ϕ(x) · ϕ(x) · w ≠ 0_{W} .

Also, g is symmetric:

(g(v_{1}))(v_{2}) = (ϕ(v_{1}) · k_{UW} (ϕ ⊗ w))(v_{2}) = ϕ(v_{1}) · ϕ(v_{2}) · w,
(g(v_{2}))(v_{1}) = (ϕ(v_{2}) · k_{UW} (ϕ ⊗ w))(v_{1}) = ϕ(v_{2}) · ϕ(v_{1}) · w.

𝔮 = 𝔣_{UW}^{−1} (g) is the quadratic form 𝔮(v) = ϕ(v) · ϕ(v) · w, and again using v = x, 𝔮 ≠ 0_{𝔔 (U;W)}.

However, 𝔱_{UU}^{W}(H) : 𝔮 ↦ 𝔮 ∘ H, and for any u ∈ U,

(𝔮 ∘ H)(u) = 𝔮(H(u)) = ϕ(H(u)) · ϕ(H(u)) · w = 0 · 0 · w = 0_{W} .

So 𝔮 ∘ H = 0_{𝔔 (U;W)}, and 𝔱_{UU}^{W}(H) is not one-to-one.

4.5.2. Algebras.

Definition 4.58. A vector space V together with a V -valued bilinear form h : V → End(V) is an \underline{algebra} (V, h).

Definition 4.59. For any algebra (V, h) with V finite-dimensional, the canonical metric (k^{∗})^{−1} ∘ e on End(V) from Example 3.143 (and Equation (3.15)) pulls back by h to give a scalar bilinear form on V, the \underline{Cartan-Killing} form

κ = h^{∗} ∘ (k^{∗})^{−1} ∘ e ∘ h : v ↦ κ(v) : u ↦ T r_{V} ((h(v)) ∘ (h(u))).

The CK form κ is symmetric by Lemma 4.17 (or Lemma 3.8 or Lemma 2.6), but is not necessarily a metric on V.

Theorem 4.60. Given an algebra (V, h), suppose Q : U → V satisfies, for any u ∈ U, v ∈ V,

(4.19)  (h(Q(u)))(v) ∈ Q(U).

If Q is one-to-one, then for any left inverse of Q, P : V ⇝ U,

(4.20)  h_{1} : U → End(U) : u ↦ P ∘ (h(Q(u))) ∘ Q

defines an algebra (U, h_{1}), and h_{1} does not depend on the choice of P. If, further, V is finite-dimensional, with CK form κ, then the CK form κ_{1} of (U, h_{1}) is the pullback of κ by Q.

Proof. Using P ∘ Q = Id_{U} and the property (4.19), for any u ∈ U and v ∈ V,

(h(Q(u)))(v) ∈ Q(U)
⟹ Q(P ((h(Q(u)))(v))) = (h(Q(u)))(v),
(4.21)  ⟹ Q ∘ P ∘ (h(Q(u))) = h(Q(u)).

Note that if P′ : V ⇝ U is any other (not necessarily linear) left inverse of Q, then composing P′ with both sides of (4.21) gives, for all u ∈ U:

P ∘ (h(Q(u))) = P′ ∘ (h(Q(u))),

so the expression (4.20) does not depend on the choice of P and defines h_{1} uniquely. Further, because h(Q(u)) ∈ End(V) has image contained in Q(U) by (4.19), the composite P ∘ (h(Q(u))) : V → U is linear by Exercise 0.45, which justifies the use of End(U) as the target space in (4.20). It remains to check that h_{1} is linear; for u_{1}, u_{2}, u_{3} ∈ U,

h_{1}(u_{1} + u_{2}) = P ∘ (h(Q(u_{1} + u_{2}))) ∘ Q
= P ∘ (h(Q(u_{1})) + h(Q(u_{2}))) ∘ Q :
u_{3} ↦ P ((h(Q(u_{1})))(Q(u_{3})) + (h(Q(u_{2})))(Q(u_{3})))
(4.22)  = P ((h(Q(u_{1})))(Q(u_{3}))) + P ((h(Q(u_{2})))(Q(u_{3}))),
= (h_{1}(u_{1}) + h_{1}(u_{2}))(u_{3}).

where step (4.22) is from Equation (0.1) in Exercise 0.45. The scaling property for h_{1} similarly follows from Exercise 0.45. The conclusion is that if Q : U → V is one-to-one, then some left inverse exists (as in Exercise 0.43) and (U, h_{1}) is an algebra as claimed. If P is a linear left inverse, then the expression (4.20) can be denoted h_{1} = Hom(Q, P) ∘ h ∘ Q.

For finite-dimensional V, U is also finite-dimensional (Exercise 0.44). The CK form on U can be computed for u_{1}, u_{2} ∈ U, using the linearity of the composites P ∘ (h(Q(u_{1}))) and P ∘ (h(Q(u_{2}))) so that Lemma 2.6 applies in step (4.23), and using Equation (4.21) in step (4.24):

(κ_{1}(u_{1}))(u_{2}) = T r_{U}((P ∘ (h(Q(u_{1}))) ∘ Q) ∘ (P ∘ (h(Q(u_{2}))) ∘ Q))
(4.23)  = T r_{V} ((Q ∘ P ∘ (h(Q(u_{1})))) ∘ (Q ∘ P ∘ (h(Q(u_{2})))))
(4.24)  = T r_{V} ((h(Q(u_{1}))) ∘ (h(Q(u_{2}))))
= ((Q^{∗} ∘ κ ∘ Q)(u_{1}))(u_{2}).

In particular, κ_{1} = Q^{∗} ∘ κ ∘ Q also does not depend on P.

Remark 4.61. The above property (4.19) represents the notion of an \underline{ideal} of the algebra (V, h). The next Theorem describes an algebra which is a direct sum of ideals.

Theorem 4.62. Given an algebra (V, h), suppose there is a direct sum V = U_{1}⊕ U_{2}, with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}). The following are equivalent.

(1) h respects the direct sums

U_{1} ⊕ U_{2} → Hom(V, U_{1}) ⊕ Hom(V, U_{2}).

(2) For both i = 1, 2, and all u ∈ U_{i}, v ∈ V,

(h(Q_{i}(u)))(v) ∈ Q_{i}(U_{i}).

Proof. In (1), the direct sum is as in Example 1.78, so the assumption is

h ∘ Q_{i} ∘ P_{i} = Hom(Id_{V} , Q_{i}) ∘ Hom(Id_{V} , P_{i}) ∘ h.

For any u ∈ U_{i}, v ∈ V,

h(Q_{i}(u)) = h(Q_{i}(P_{i}(Q_{i}(u))))
= Q_{i} ∘ P_{i} ∘ (h(Q_{i}(u))) :
v ↦ Q_{i}(P_{i}((h(Q_{i}(u)))(v))) ∈ Q_{i}(U_{i}),

so (1) ⟹ (2). Conversely, assuming (2), for u ∈ U_{i}, v ∈ V, and different indices i ≠ I,

(h(Q_{i}(u)))(v) ∈ Q_{i}(U_{i})
⟹ (h(Q_{i}(u)))(v) = Q_{i}(P_{i}((h(Q_{i}(u)))(v)))
⟹ h(Q_{i}(u)) = Q_{i} ∘ P_{i} ∘ (h(Q_{i}(u)))
⟹ h ∘ Q_{i} = Hom(Id_{V} , Q_{i} ∘ P_{i}) ∘ h ∘ Q_{i}
⟹ Hom(Id_{V} , P_{I}) ∘ h ∘ Q_{i} = Hom(Id_{V} , P_{I}) ∘ Hom(Id_{V} , Q_{i} ∘ P_{i}) ∘ h ∘ Q_{i}
= 0_{Hom(U_{i},Hom(V,U_{I}))},

so by Lemma 1.83, h respects the direct sums as in (1).

For a direct sum as in Theorem 4.62, Theorem 4.60 applies, so that each U_{i} has an algebra structure (U_{i}, h_{i}), and if V is finite-dimensional, then each (U_{i}, h_{i}) has CK form κ_{i} = Q_{i}^{∗} ∘ κ ∘ Q_{i}.

Theorem 4.63. For an algebra (V, h), the following are equivalent.

(1) For all u, v, w ∈ V,

(h(u))((h(v))(w)) = (h((h(u))(v)))(w).

(2) For any u ∈ V, this diagram is commutative.

\xymatrix{V \ar[r]^{h(u)} \ar[d]^{h} & V \ar[d]^{h} \\ End(V) \ar[r]^{Hom(Id_{V} ,h(u))} & End(V)}

Proof. (1) is equivalent to: for all u, v,

(h(u)) ∘ (h(v)) = h((h(u))(v)),

which is equivalent to Hom(Id_{V} , h(u)) ∘ h = h ∘ (h(u)) for all u, which is (2).

Definition 4.64. An algebra (V, h) satisfying either equivalent property from Theorem 4.63 is an \underline{associative algebra}.

Example 4.65. The generalized transpose from Definition 1.7 and Example 1.49,

t_{V V}^{V} : End(V) → End(End(V)) : A ↦ Hom(A, Id_{V}) : B ↦ B ∘ A,

defines an associative algebra (End(V), t_{V V}^{V}). Using Corollary 2.37,

(κ(A))(B) = T r_{End(V)}((t_{V V}^{V} (A)) ∘ (t_{V V}^{V} (B)))
= T r_{End(V)}(Hom(A, Id_{V}) ∘ Hom(B, Id_{V}))
= T r_{End(V)}(Hom(B ∘ A, Id_{V}))
= T r_{V} (B ∘ A) · T r_{V} (Id_{V}),

so the form κ for the algebra (End(V), t_{V V}^{V}) is a scalar multiple of the canonical metric from Example 3.143.

Definition 4.66. For any vector space V, define the linear map ad,

ad : End(V) → End(End(V)) :
A ↦ Hom(Id_{V} , A) − Hom(A, Id_{V}) :
B ↦ A ∘ B − B ∘ A.

Theorem 4.67. For an algebra (V, h), if h is antisymmetric then the following are equivalent.

(1) For all u, v, w ∈ V,

(h((h(v))(w)))(u) + (h((h(w))(u)))(v) + (h((h(u))(v)))(w) = 0_{V} .

(2) For any v ∈ V, this diagram is commutative.

\xymatrix{V \ar[r]^{h(v)} \ar[d]^{h} & V \ar[d]^{h} \\ End(V) \ar[r]^{ad(h(v))} & End(V)}

Proof. The first property is the \underline{Jacobi identity} for h. In (2), the map ad : End(V) → End(End(V)) is as in Definition 4.66. Using the antisymmetric property,

(1) is equivalent to

(h((h(v))(w)))(u) = (h(v))((h(w))(u)) − (h(w))((h(v))(u))

for all u, v, w, which is equivalent to, for all v, w,

h((h(v))(w)) = (h(v)) ∘ (h(w)) − (h(w)) ∘ (h(v))
= (ad(h(v)))(h(w)).

This is equivalent to h ∘ (h(v)) = (ad(h(v))) ∘ h for all v, which is (2).

Definition 4.68. An algebra (V, h) with h antisymmetric and satisfying either equivalent property from Theorem 4.67 is a \underline{Lie algebra}.

Exercise 4.69. If \frac{1}{2} ∈ 𝕂 and h : V → End(V) satisfies (1) from Theorem 4.63, then its antisymmetric part \frac{1}{2}(h − T_{V;V} (h)) from (4.7) satisfies (1) from Theorem 4.67. Similarly, h − T_{V;V} (h) satisfies the Jacobi identity even without assuming \frac{1}{2} ∈ 𝕂. So for any associative algebra (V, h), there is a Lie algebra (V, h−T_{V;V} (h)).

Example 4.70. (End(V), ad) is a Lie algebra. This uses the construction of Exercise 4.69 applied to (End(V), t_{V V}^{V}) from Example 4.65, although with the opposite sign, so that ad = T_{V;V} (t_{V V}^{V}) − t_{V V}^{V}. For finite-dimensional V, the form κ is the pullback of the canonical metric on End(End(V)) (from Equation (3.16)) by ad (or by its opposite, −ad),

κ = (±ad)^{∗} ∘ (k_{End(V),End(V)}^{∗})^{−1} ∘ e_{End(V),End(V)} ∘ (±ad).

Using Equation (3.15), Lemma 2.6, and Corollary 2.37,

(κ(A))(B) = T r_{End(V)}((ad(A)) ∘ (ad(B))) (4.25)

= T r_{End(V)}((Hom(Id_{V} , A) − Hom(A, Id_{V})) ∘ (Hom(Id_{V} , B) − Hom(B, Id_{V})))
= T r_{End(V)}(Hom(Id_{V} , A ∘ B) − Hom(A, B) − Hom(B, A) + Hom(B∘ A, Id_{V}))
= 2 · T r_{V} (Id_{V}) · T r_{V} (A ∘ B) − 2 · T r_{V} (A) · T r_{V} (B).

Example 4.71. Direct sums of ideals as in Theorem 4.62 are considered in the Lie algebra case by [Humphreys] §II.5. If V is finite-dimensional and T r_{V} (Id_{V}) ≠ 0, then Theorem 4.60 applies to the Lie algebra (End(V), ad) and the direct sum End(V) = 𝕂 ⊕ End_{0}(V) from Example 2.9. The corresponding CK forms are κ_{1} = 0_{Hom(𝕂,𝕂^{∗})} and κ_{2} = Q_{2}^{∗} ∘ κ ∘ Q_{2}, so that for trace 0 elements A, B ∈ End_{0}(V), the pullback of (4.25) by the inclusion Q_{2} gives (κ_{2}(A))(B) = 2·T r_{V} (Id_{V})·T r_{V} (A∘B).

Example 4.72. For a vector space U, with \frac{1}{2} ∈ 𝕂, and a metric g : U → U^{∗}, recall from Definition 3.116 the direct sum on End(U) produced by the involution Hom(g, g^{−1})∘t, defining subspaces of self-adjoint and skew-adjoint endomorphisms, temporarily denoted here by:

End(U) = 𝔰𝔞 ⊕ 𝔰𝔬
= {A = g^{−1} ∘ A^{∗} ∘ g} ⊕ {A = −g^{−1} ∘ A^{∗} ∘ g}.

Let (P_{1}, P_{2}), (Q_{1}, Q_{2}) denote the projection and inclusions for the direct sum. Then with Hom(Q_{2}, P_{2}) ∘ ad ∘ Q_{2} : 𝔰𝔬 → End(𝔰𝔬) as in (4.20), (𝔰𝔬, Hom(Q_{2}, P_{2}) ∘ ad ∘ Q_{2}) is a Lie algebra. It is a subalgebra, but not an ideal, of (End(U), ad), so Theorem 4.60 and Theorem 4.62 do not apply.

Definition 4.73. An algebra (V, h) is a \underline{one-sided division algebra} means that for every v ≠ 0_{V}, h(v) is invertible. (V, h) is a \underline{two-sided division algebra} means that for every v ≠ 0_{V}, both h(v) and (T_{V;V} (h))(v) are invertible.

4.5.3. Curvature tensors.

Example 4.74. Consider V with a metric g, and an End(V)-valued bilinear form R ∈ Hom(V, Hom(V, End(V))).

If \frac{1}{2} ∈ 𝕂 and R satisfies

(4.26)  T_{V;End(V)}(R) = −R

so that R ∈ Alt(V; End(V)), then by Corollary 4.32,

T r_{g;End(V)}(R) = 0_{End(V)}.
If \frac{1}{2} ∈ 𝕂 and R satisfies
(4.27)  Hom(Id_{V} , Hom(Id_{V} , Hom(g, g^{−1}) ∘ t))(R) = −R

so that for any u, v ∈ V,

(R(u))(v) = −g^{−1} ∘ ((R(u))(v))^{∗} ∘ g ∈ End(V),

then (R(u))(v) is skew-adjoint with respect to g and, as in Exercise 3.119,

Hom(Id_{V} , Hom(Id_{V} , T r_{V}))(R) : u ↦
((Hom(Id_{V} , T r_{V})) ∘ R)(u) : v ↦ (T r_{V} ∘ (R(u)))(v)
= T r_{V} ((R(u))(v)) = 0
(4.28) ⟹ Hom(Id_{V} , Hom(Id_{V} , T r_{V}))(R) = 0_{Hom(V,V^{∗})}.

There is another trace that is not necessarily zero even under both of the above conditions. Define:

Ric = (Hom(Id_{V} , Hom(Id_{V} , T r_{V})) ∘ Hom(Id_{V} , T_{V;V})) (R) ∈ Hom(V, V^{∗}).

If T r_{V} (Id_{V}) ≠ 0, then Theorem 3.35 and Theorem 3.53 apply — there is a canonical, orthogonal decomposition:

(4.29)  Ric = \frac{T r_{g}(Ric)}{T r_{V} (Id_{V})} · g + (Ric − \frac{T r_{g}(Ric)}{T r_{V} (Id_{V})} · g) .

Remark 4.75. For a smooth manifold where V is a tangent vector space over 𝕂 = ℝ and g is a pseudo-Riemannian metric, the linear algebra properties of the Riemann curvature tensor R_{qkl}^{i} at a point are modeled by a form R as in Example 4.74, with the symmetries (4.26), (4.27), and (4.30). Its trace Ric is the Ricci curvature tensor, and T r_{g}(Ric) from (4.29) is the scalar curvature. See [DFN] §30.

Remark 4.76. The second term from (4.29) is the \underline{trace-free Ricci tensor}. There are other interesting linear combinations of Ric and g, including the tracereversed Ricci tensor,

Ric − 2 · \frac{T r_{g}(Ric)}{T r_{V} (Id_{V})} · g,

and the Einstein tensor,

Ric − \frac{1}{2} · T r_{g}(Ric) · g.

Proposition 4.77. If \frac{1}{2} ∈ 𝕂 and R ∈ Hom(V, Hom(V, End(V))) satisfies (4.26) and (4.28), and additionally has the property

(4.30) (T_{V;End(V)} ∘ Hom(Id_{V} , T_{V;V}) + Hom(Id_{V} , T_{V;V}) ∘ T_{V;End(V)})(R) = −R,

then Ric is symmetric.

Proof. Using Lemma 4.6, (4.30), (4.28), and (4.26),

T_{V} (Ric) = (T_{V;𝕂} ∘ Hom(Id_{V} , Hom(Id_{V} , T r_{V})) ∘ Hom(Id_{V} , T_{V;V}))(R)
= (Hom(Id_{V} , Hom(Id_{V} , T r_{V})) ∘ T_{V;End(V)} ∘ Hom(Id_{V} , T_{V;V}))(R)
= −Hom(Id_{V} , Hom(Id_{V} , T r_{V}))(R)

−(Hom(Id_{V} , Hom(Id_{V} , T r_{V})) ∘ Hom(Id_{V} , T_{V;V}) ∘ T_{V;End(V)})(R)

= −0_{Hom(V,V^{∗})}
−(Hom(Id_{V} , Hom(Id_{V} , T r_{V})) ∘ Hom(Id_{V} , T_{V;V}))(−R) = Ric.

If, further, T r_{V} (Id_{V}) ≠ 0, then it follows that the second, trace-free term in (4.29) is also symmetric.

Remark 4.78. The lowered-index curvature tensor R_{iqkl} is modeled by the multilinear form R′ in the following Example 4.79. Then Proposition 4.80 demonstrates the symmetry property R_{iqkl} = R_{kliq}.

Example 4.79. For V, g, and R ∈ Hom(V, Hom(V, End(V))) as in Example 4.74, define

R′ = Hom(Id_{V} , Hom(Id_{V} , Hom(Id_{V} , g)))(R) ∈ Hom(V, Hom(V, Hom(V, V^{∗}))), so that for u, v ∈ V, (R′(u))(v) = g ∘ ((R(u))(v)) : V → V^{∗}. As in Theorem 3.115 and Example 3.117, if \frac{1}{2} ∈ 𝕂 and (R(u))(v) is skew-adjoint then (R′(u))(v) is an antisymmetric form, so if R has property (4.27) then R′ satisfies

(4.31)  Hom(Id_{V} , Hom(Id_{V} , T_{V}))(R′) = −R′.

By Lemma 4.6,

Hom(Id_{V} , Hom(Id_{V} , Hom(Id_{V} , g))) ∘ T_{V;End(V)}
= T_{V;Hom(V,V^{∗})} ∘ Hom(Id_{V} , Hom(Id_{V} , Hom(Id_{V} , g))),

so if R has property (4.26), then R′ satisfies

(4.32)  T_{V;Hom(V,V^{∗})}(R′) = −R′.

Similarly by Lemma 4.6,

Hom(Id_{V} , Hom(Id_{V} , Hom(Id_{V} , g))) ∘ Hom(Id_{V} , T_{V;V})
= Hom(Id_{V} , T_{V;V^{∗}}) ∘ Hom(Id_{V} , Hom(Id_{V} , Hom(Id_{V} , g))),

so if R has property (4.30), then R′ satisfies (4.33) (T_{V;Hom(V,V^{∗})} ∘ Hom(Id_{V} , T_{V;V^{∗}}) + Hom(Id_{V} , T_{V;V^{∗}}) ∘ T_{V;Hom(V,V^{∗})})(R′) = −R′.

Proposition 4.80. If \frac{1}{2} ∈ 𝕂 and R′ ∈ Hom(V, Hom(V, Hom(V, V^{∗}))) satisfies (4.31), (4.32), and (4.33), then R′ is a fixed point of the involution (4.34) Hom(Id_{V} , T_{V;V^{∗}}) ∘ T_{V;Hom(V,V^{∗})} ∘ Hom(Id_{V} , Hom(Id_{V} , T_{V})) ∘ Hom(Id_{V} , T_{V;V^{∗}}).

Proof. Temporarily denote the involutions:

a_{1} = T_{V;Hom(V,V^{∗})}
a_{2} = Hom(Id_{V} , T_{V;V^{∗}})
a_{3} = Hom(Id_{V} , Hom(Id_{V} , T_{V})).

Properties (4.31) and (4.32) then can be stated:

(4.35)  a_{1}(R′) = a_{3}(R′) = −R′.

By Lemma 4.6, a_{1} ∘ a_{3} = a_{3} ∘ a_{1}, and this is enough to show that the composite a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2} from (4.34) is an involution. Lemma 4.12 gives the relations:

a_{2} ∘ a_{1} ∘ a_{2} = a_{1} ∘ a_{2} ∘ a_{1}
a_{3} ∘ a_{2} ∘ a_{3} = a_{2} ∘ a_{3} ∘ a_{2}.

Starting with property (4.33), applying a_{3} to both sides, and then using (4.35) gives:

(4.36)  (a_{1} ∘ a_{2} + a_{2} ∘ a_{1})(R′) = −R′
⟹ (a_{3} ∘ a_{1} ∘ a_{2} + a_{3} ∘ a_{2} ∘ a_{1})(R′) = a_{3}(−R′)
(a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2} ∘ a_{3} + a_{2} ∘ a_{3} ∘ a_{2} ∘ a_{3} ∘ a_{1})(R′) = R′
(4.37)  ⟹ (a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2} + a_{2} ∘ a_{3} ∘ a_{2} ∘ a_{3})(R′) = −R′.

Applying a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2} to both sides of (4.36) gives:

(a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2} ∘ a_{1} ∘ a_{2})(R′)
+(a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2} ∘ a_{2} ∘ a_{1})(R′) = (a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2})(−R′)
(4.38)  (a_{2} ∘ a_{3} ∘ a_{2} ∘ a_{1} + a_{2} ∘ a_{1})(R′) = −(a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2})(R′).

Applying a_{1} to both sides of (4.38) gives:

(a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2} ∘ a_{1} + a_{1} ∘ a_{2} ∘ a_{1})(R′) = −(a_{1} ∘ a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2})(R′)
(a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2} + a_{1} ∘ a_{2})(−R′) = −(a_{2} ∘ a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2})(R′)
(4.39) ⟹ (a_{1} ∘ a_{2} ∘ a_{3} ∘ a_{2} + a_{1} ∘ a_{2})(R′) = −(a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2})(R′).

Adding (4.36) and (4.37) and subtracting (4.38) and (4.39), there are cancellations using (4.35) again, to get:

0_{Hom(V,Hom(V,Hom(V,V^{∗})))} = −2 · R′ + 2 · (a_{2} ∘ a_{1} ∘ a_{3} ∘ a_{2})(R′),

which proves the claim.

4.5.4. Partially symmetric forms. A map A ⊗ V ⊗ V → F is called a “trilinear F -form” in [EHM], and it is “partially symmetric” means that it is invariant under switching of the V factors. Such forms, of course, lie in the scope of these notes, and it will also be convenient to consider maps of the form

V ⊗ U → Hom(V, W),

as in the vector valued forms of Sections 4.2 and 4.3, but with the domain twisted by U. The two notions are related by a canonical map, as already seen in the Proof of Lemma 4.12.

Notation 4.81. For arbitrary V, U, W, X, define

(4.40)  q : Hom(X ⊗ U, Hom(V, W)) → Hom(V ⊗ X ⊗ U, W)

so that for G : X ⊗ U → Hom(V, W), v ∈ V, x ∈ X, and u ∈ U,

q(G) : v ⊗ x ⊗ u ↦ (G(x ⊗ u))(v).

The order of the factors is different from that in Definition 1.44, but such maps will still have “q” labels (in analogy with the variations on n from Notation 1.40).

In the following Lemma, T_{V} is the transpose map from Definition 3.2, and q_{1} is the q map from (4.40) in the case X = V.

Lemma 4.82. The following diagram is commutative.

\xymatrix{Hom(V,V^{∗})⊗Hom(U,W) \ar[d]_{j} \ar[r]^{[T_{V}⊗Id_{Hom(U,W)}]} & Hom(V,V^{∗})⊗Hom(U,W) \ar[d]^{j} \\ Hom(V ⊗ U, V^{∗} ⊗ W) \ar[d]_{Hom(Id_{V ⊗U} ,k_{V W})} & Hom(V ⊗ U, V^{∗} ⊗ W) \ar[d]^{Hom(Id_{V ⊗U} ,k_{V W})} \\ Hom(V ⊗ U, Hom(V, W)) \ar[d]_{q_{1}} & Hom(V ⊗ U, Hom(V, W)) \ar[d]^{q_{1}} \\ Hom(V ⊗ V ⊗ U, W) \ar[r]^{Hom([s⊗Id_{U}],Id_{W})} & Hom(V ⊗ V ⊗ U, W)}

Proof. Without stating all the details, the upper part of the diagram is analogous to the diagram from Lemma 4.5, and the lower part is analogous to the diagram from Lemma 4.1.

Definition 4.83. Define T_{V;U,W} ∈ End(Hom(V ⊗ U, Hom(V, W))) by

T_{V;U,W} = q_{1}^{−1} ∘ Hom([s ⊗ Id_{U}], Id_{W}) ∘ q_{1}.

With this construction, the T_{V;U,W} maps are analogous to, but not a special case of, the maps T_{V_{1},V_{2};W} from Lemma 4.1 and Definition 4.2. If V is finitedimensional, then j and k in the above diagram are invertible, and by the Lemma,

T_{V;U,W} = Hom(Id_{V ⊗U}, k_{V W}) ∘ j ∘ [T_{V} ⊗ Id_{Hom(U,W)}] ∘ j^{−1} ∘ Hom(Id_{V ⊗U}, k_{V W}^{−1} ).

As in Section 4.2, T_{V;U,W} is an involution, and if \frac{1}{2} ∈ 𝕂, then it produces a direct sum structure on Hom(V ⊗ U, Hom(V, W)), by Lemma 1.114. The other two involutions in the above diagram also produce direct sums, and by Lemma 1.120, the maps q_{1} and Hom(Id_{V ⊗U}, k_{V W}) ∘ j respect these direct sums, although the comments about Example 1.139 in Remark 4.14 apply here also.

Exercise 4.84. With respect to induced metrics, T_{V;U,W} is an isometry, and if \frac{1}{2} ∈ 𝕂, then it produces an orthogonal direct sum.

Hint. Use Definition 4.83 to show T_{V;U,W} is a composition of isometries. Then Lemma 3.55 applies, as in Theorem 4.39.

Definition 4.85. A map G : V ⊗ U → Hom(V, W) is \underline{partially symmetric} means: T_{V;U,W} (G) = G. More generally, a map G : X ⊗ U → Hom(V, W) is partially symmetric with respect to a map H : V → X means that G ∘ [H ⊗ Id_{U}] : V ⊗ U → Hom(V, W) is partially symmetric.

Lemma 4.86. ([EHM]) For any V, U, W, X, and G : X ⊗ U → Hom(V, W), the following diagram is commutative.

\xymatrix{𝕂 ⊗ X ⊗ U \ar[d]^{[Q_{1}^{1}⊗Id_{X⊗U}]} \ar[r]^{l} & X ⊗ U \ar[dd]^{G} \\ End(V) ⊗ X ⊗ U \ar[d]^{n} \\ Hom(V, V ⊗ X ⊗ U) \ar[r]^{Hom(Id_{V} ,q(G))} & Hom(V, W) \\ V^{∗} ⊗ V ⊗ X ⊗ U \ar[r]^{[Id_{V ^{∗}} ⊗(q(G))]} \ar[u]^{k_{V,V ⊗X⊗U}} & V^{∗} ⊗ W \ar[u]^{k_{V W}}}

Proof. The lower square uses Lemma 1.58. In the upper square, the maps are n as in Definition 1.39, and an inclusion Q_{1}^{1} : λ → λ · Id_{V} as in Example 2.9 and Equation (2.4).

λ ⊗ x ⊗ u ↦ (Hom(Id_{V} , q(G)) ∘ n ∘ [Q_{1}^{1} ⊗ Id_{X⊗U}])(λ ⊗ x ⊗ u)
= (q(G)) ∘ (n(λ · Id_{V} ⊗ x ⊗ u)) :
v ↦ (q(G))(λ · v ⊗ x ⊗ u)
= λ · (G(x ⊗ u))(v) = ((G ∘ l)(λ ⊗ x ⊗ u))(v).

Theorem 4.87. For any spaces U, V, W, X, Y, and any maps G : X ⊗ U → Hom(V, W), M : Y → X ⊗ U, the following are equivalent.

(q(G)) ∘ [Id_{V} ⊗ M] = 0_{Hom(V ⊗Y,W)}
⟺ G ∘ M = 0_{Hom(Y,Hom(V,W))}.

Proof. Let q be the map from Notation 4.81, and let q_{2} be another such map in the following diagram.

\xymatrix{Hom(X ⊗ U, Hom(V, W)) \ar[r]^{q} \ar[d]_{Hom(M,Id_{Hom(V,W)})} & Hom(V ⊗ X ⊗ U, W) \ar[d]^{Hom([Id_{V} ⊗M],Id_{W})} \\ Hom(Y, Hom(V, W)) \ar[r]^{q_{2}} & Hom(V ⊗ Y, W)}

The diagram is commutative by Lemma 1.47. So, q_{2}(G ∘ M) = (q(G)) ∘ [Id_{V} ⊗ M]. Since q_{2} is invertible (Lemma 1.45), G ∘ M is zero if and only if its output under q_{2} is also zero.

Example 4.88. Suppose there is some direct sum V ⊗ X ⊗ U = W_{1} ⊕ W_{2}, with projections P_{i} : V ⊗ X ⊗ U → W_{i}. Then, for q_{i} : Hom(X ⊗ U, Hom(V, W_{i})) → Hom(V ⊗ X ⊗ U, W_{i}) and M : Y → X ⊗ U,

P_{i} ∘ [Id_{V} ⊗ M] = 0_{Hom(V ⊗Y,W_{i})} ⟺ (q_{i}^{−1}(P_{i})) ∘ M = 0_{Hom(Y,Hom(V,W_{i}))}.

The following Theorem uses the the switching involution s as in Lemma 4.82, and the direct sum V ⊗ V = S^{2}V ⊕ Λ^{2}V produced by s as in Example 1.118, with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}).

Theorem 4.89. Let H : V → X, and suppose there is some direct sum

V ⊗ X = Z_{1} ⊕ Z_{2}

with operators P′_{i}, Q′_{i}, such that [Id_{V} ⊗ H] : V ⊗ V → V ⊗ X respects the direct sums and P′_{2} ∘[Id_{V} ⊗H]∘Q_{2} : Λ^{2}V → Z_{2} is invertible. If G : X ⊗U → Hom(V, W) is partially symmetric with respect to H, then

q(G) = (q(G)) ∘ [(Q′_{1} ∘ P′_{1}) ⊗ Id_{U}].

Proof. The following diagram is commutative by Lemma 1.47, where q_{1} is as in Lemma 4.82.

\xymatrix{Hom(X ⊗ U, Hom(V, W)) \ar[r]^{q} \ar[d]_{Hom([H⊗Id_{U}],Id_{Hom(V,W)})} & Hom(V ⊗ X ⊗ U, W) \ar[d]^{Hom([Id_{V} ⊗[H⊗Id_{U}]],Id_{W})} \\ Hom(V ⊗ U, Hom(V, W)) \ar[r]^{q_{1}} & Hom(V ⊗ V ⊗ U, W)}

By the assumption about respecting the direct sum, P′_{I} ∘ [Id_{V} ⊗ H] ∘ Q_{i} is zero for i ≠ I. Let P′′_{2} denote the projection \frac{1}{2} · (Id_{Hom(V ⊗U,Hom(V,W))} − T_{V;U,W}), so that if

188 4. VECTOR VALUED BILINEAR FORMS

G is partially symmetric with respect to H, then

0_{Hom(Λ^{2}V ⊗U,W)} = (q_{1}(P′′_{2}(G ∘ [H ⊗ Id_{U}]))) ∘ [Q_{2} ⊗ Id_{U}]
= (\frac{1}{2} · q_{1}(G ∘ [H ⊗ Id_{U}])) ∘ [Q_{2} ⊗ Id_{U}]

−(\frac{1}{2} · q_{1}(G ∘ [H ⊗ Id_{U}])) ∘ [s ⊗ Id_{U}] ∘ [Q_{2} ⊗ Id_{U}]

= (q_{1}(G ∘ [H ⊗ Id_{U}])) ∘ [Q_{2} ⊗ Id_{U}] ∘ [P_{2} ⊗ Id_{U}] ∘ [Q_{2} ⊗ Id_{U}]
= (q(G)) ∘ [Id_{V} ⊗ [H ⊗ Id_{U}]] ∘ [Q_{2} ⊗ Id_{U}]
= (q(G)) ∘ ([Q′_{1} ⊗ Id_{U}] ∘ [P′_{1} ⊗ Id_{U}] + [Q′_{2} ⊗ Id_{U}] ∘ [P′_{2} ⊗ Id_{U}])

∘[[Id_{V} ⊗ H] ⊗ Id_{U}] ∘ [Q_{2} ⊗ Id_{U}]

= (q(G)) ∘ [Q′_{2}⊗Id_{U}] ∘ [P′_{2}⊗Id_{U}] ∘ [[Id_{V} ⊗H]⊗Id_{U}] ∘ [Q_{2}⊗Id_{U}].

Then, since [P′_{2} ⊗ Id_{U}] ∘ [[Id_{V} ⊗ H] ⊗ Id_{U}] ∘ [Q_{2} ⊗ Id_{U}] is invertible,

0_{Hom(Z_{2}⊗U,W)} = (q(G)) ∘ [Q′_{2} ⊗ Id_{U}],

and it follows that q(G) = (q(G)) ∘ [(Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2}) ⊗ Id_{U}] = (q(G)) ∘ [(Q′_{1} ∘ P′_{1}) ⊗ Id_{U}].

It follows from the previous two Theorems that if M : Y → X ⊗ U and G is partially symmetric with respect to H, then

q_{2}(G ∘ M) = (q(G)) ∘ [(Q′_{1} ∘ P′_{1}) ⊗ Id_{U}] ∘ [Id_{V} ⊗ M]. Remark 4.90. The map H : V → X could be an inclusion of a vector subspace,

in which case the above Z_{1} corresponds to the space denoted by H.V in [EHM].

Big Exercise 4.91. Given a metric g on V, there exists a trace operator T r_{g;U,W} : Hom(V ⊗ U, Hom(V, W)) → Hom(U, W)

having nice properties which follow as corollaries of the results in Section 2.2.

CHAPTER 5 Complex Structures

At this point we abandon the general field 𝕂 and work exclusively with real number scalars and vector spaces over the field ℝ. Some of the objects could be considered vector spaces over the field of complex numbers, but in this Chapter, complex numbers will not be used as scalars or for any other purpose. The objects will instead be real vector spaces paired with some additional structure, and the maps are all ℝ-linear, although some of the ℝ-linear maps will respect the additional structure.

5.1. Complex Structure Operators

Definition 5.1. Given a real vector space V, an endomorphism J ∈ End(V) is a \underline{complex structure operator} means: J ∘ J = −Id_{V}.

Notation 5.2. A complex structure operator is more briefly called a \underline{CSO}. Sometimes a pair (V, J) will be denoted by a matching boldface letter, V. Expressions such as v ∈ V, A : U → V, etc., refer to the underlying real space v ∈ V, A : U → V, etc.

Example 5.3. Given V = (V, J_{V}) and another vector space U, [Id_{U} ⊗ J_{V}] ∈ End(U ⊗V) is a canonical CSO on U ⊗V, so we may denote U ⊗V = (U ⊗V, [Id_{U} ⊗ J_{V}]). Similarly, denote V ⊗ U = (V ⊗ U, [J_{V} ⊗ Id_{U}]).

Example 5.4. Given a vector space V with CSO J_{V} and another vector space U, Hom(Id_{U}, J_{V}) : A ↦ J_{V} ∘ A is a canonical CSO on Hom(U, V), so we may denote Hom(U, V) = (Hom(U, V), Hom(Id_{U}, J_{V})). Similarly, denote Hom(V, U) = (Hom(V, U), Hom(J_{V} , Id_{U})).

Example 5.5. Given V, a CSO J induces a CSO J^{∗} = Hom(J, Id_{ℝ}) on V^{∗} = Hom(V, ℝ).

Example 5.6. Given V, a CSO J ∈ End(V), and any involution N that commutes with J (i.e., N ∈ End(V) such that N ∘ N = Id_{V} and N ∘ J = J ∘ N), N ∘ J is a CSO.

Example 5.7. Given V ≠ {0_{V} }, any CSO J ∈ End(V) is not unique, since −J is a different CSO. This is the N = −Id_{V} case from Example 5.6.

Example 5.8. Given V = V_{1} ⊕V_{2}, suppose there is an invertible map A : V_{2} → V_{1}, as in (3) from Theorem 1.130. Then, V also admits a CSO,

(5.1)  J = Q_{2} ∘ A^{−1} ∘ P_{1} − Q_{1} ∘ A ∘ P_{2},

and its opposite, −J. In the special case (1) where V = U ⊕ U, A = Id_{V}, the above CSO is J = Q_{2} ∘ P_{1} − Q_{1} ∘ P_{2}. In the equivalent situation, (8) from Theorem 1.130, where there exist anticommuting involutions K_{1} and K_{2} on V, the composite K_{1} ∘ K_{2} and its opposite are CSOs on V. Using K_{1} and K_{2} to define a direct sum and a map A as in (1.18), the above construction (5.1) gives the same pair of CSOs: {±K_{1} ∘ K_{2}} = {±J}.

Lemma 5.9. Given V with CSO J and v ∈ V, if v ≠ 0_{V}, then the ordered list (v, J(v)) is linearly independent.

Proof. Linear independence as in Definition 0.27 refers to the scalar field ℝ in this Chapter. If J(v) = α · v for some α ∈ ℝ, then J(J(v)) = (−1) · v = α^{2} · v; there are no solutions for v ≠ 0_{V} and α ∈ ℝ.

Exercise 5.10. Not every real vector space admits a CSO.

Lemma 5.11. Given V with CSO J and v_{1}, …, v_{ℓ} ∈ V, if the ordered list

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, J(v_{1}), …, J(v_{ℓ−1}))

is linearly independent, then so is the ordered list

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, J(v_{1}), …, J(v_{ℓ−1}), J(v_{ℓ})) .

Proof. The ℓ = 1 case is Lemma 5.9. For ℓ ≥ 2, suppose there are real scalars α_{1}, …, α_{ℓ}, β_{1}, …, β_{ℓ} such that

(5.2)  α_{1} · v_{1} + … + α_{ℓ} · v_{ℓ} + β_{1} · J(v_{1}) + … + β_{ℓ} · J(v_{ℓ}) = 0_{V} .

Then,

(5.3)  α_{1} · J(v_{1}) + … + α_{ℓ} · J(v_{ℓ}) − β_{1} · v_{1} − … − β_{ℓ} · v_{ℓ} = 0_{V} .

Subtracting α_{ℓ} times (5.2) minus β_{ℓ} times (5.3), the J(v_{ℓ}) terms cancel, and

(α_{1}α_{ℓ} + β_{1}β_{ℓ}) · v_{1} + … + (α_{ℓ−1}α_{ℓ} + β_{ℓ−1}β_{ℓ}) · v_{ℓ−1} + (α_{ℓ}^{2} + β_{ℓ}^{2}) · v_{ℓ}
+(α_{ℓ}β_{1} − α_{1}β_{ℓ}) · J(v_{1}) + … + (α_{ℓ}β_{ℓ−1} − α_{ℓ−1}β_{ℓ}) · J(v_{ℓ−1}) = 0_{V} .

By the linear independence of the ordered list with 2ℓ − 1 elements, α_{ℓ}^{2} + β_{ℓ}^{2} = 0 ⟹ α_{ℓ} = β_{ℓ} = 0. Then (5.2) and the independence hypothesis again (or Lemma 5.9 if ℓ = 2) imply α_{1} = … = α_{ℓ−1} = β_{1} = … = β_{ℓ−1} = 0.

Definition 5.12. Given V with CSO J, a subspace H is \underline{J-invariant} means: J(H) ⊆ H. Equivalently, because J is invertible, J(H) = H.

Lemma 5.13. Given V with CSO J, and a J-invariant subspace H of V, if H ≠ {0_{V} } and H is finite-dimensional, then H admits an ordered basis of the form

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, J(v_{1}), …, J(v_{ℓ−1}), J(v_{ℓ})) .

Proof. By hypothesis, there is some v_{1} ∈ H with v_{1} ≠ 0_{V}, and by Jinvariance, J(v) ∈ H, so by Lemma 5.9, (v_{1}, J(v_{1})) is a linearly independent ordered list of vectors in H. Suppose inductively that

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, J(v_{1}), …, J(v_{ℓ−1}), J(v_{ℓ}))

is a linearly independent ordered list of vectors in H. If the ordered list spans H, it is an ordered basis; otherwise, there is some v_{ℓ+1} ∈ H not in its span, so

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, v_{ℓ+1}, J(v_{1}), …, J(v_{ℓ−1}), J(v_{ℓ}))

is a linearly independent ordered list. J(v_{ℓ+1}) ∈ H and Lemma 5.11 applies, so

(v_{1}, …, v_{ℓ−1}, v_{ℓ}, v_{ℓ+1}, J(v_{1}), …, J(v_{ℓ−1}), J(v_{ℓ}), J(v_{ℓ+1}))

is a linearly independent ordered list of elements in H. The construction eventually terminates by Claim 0.28.

Definition 5.14. Given V with CSO J, a subspace H of V which is equal to a span of a two-element set and is also J-invariant, H = J(H) ⊆ V, will be called a \underline{J-complex line} in V.

By Lemma 5.13, a J-complex line H must be of the form span{v, J(v)} for some non-zero v ∈ H.

Lemma 5.15. Given V with CSO J, a J-complex line L, and a J-invariant subspace H ⊆ V, if there is a non-zero element v ∈ L ∩ H, then L ⊆ H. In particular, if H is a J-complex line, then L = H.

Proof. By J-invariance, {v, J(v)} ⊆ L ∩ H. By Lemma 5.9, (v, J(v)) is a linearly independent ordered list, so it is an ordered basis of L and its span is contained in H. Comment: the contrapositive can be stated: Given V with CSO J, v ∈ V, and a J-invariant subspace H, if v ∉ H, then H ∩span{v, J(v)} = {0_{V} }.

Lemma 5.16. Given V with CSO J, if L^{1}, L^{2} are distinct J-complex lines in V, then span(L^{1} ∪ L^{2}) has an ordered basis with 4 elements. In particular, a subspace H ⊆ V that does not have 4 elements forming a linearly independent list can contain at most one J-complex line.

Proof. Suppose L_{1} is a J-complex line in V, with L_{1} = span{v, J(v)}. If L_{2} is a J-complex line with L_{2} ≠ L_{1}, then L_{2} ⊈ L_{1}, so there is some u ∈ L_{2} \ L_{1}, and (v, J(v), u) is a linearly independent list. Because L_{2} is J-invariant, J(u) ∈ L_{2}, so by Lemma 5.11, (v, J(v), u, J(u)) is a linearly independent ordered list of elements of L^{1} ∪ L^{2}, and an ordered basis of span(L^{1} ∪ L^{2}).

Lemma 5.17. Given V with CSO J, and a subspace H of V, if H = {0_{V} } or H has an ordered basis of the form (u_{1}, J(u_{1}), …, u_{ν}, J(u_{ν})), then either H = V or there exists a subspace U of V such that U is J-invariant, H ⊆ U, and U admits an ordered basis with 2(ν + 1) elements.

Proof. This is trivial for H = {0_{V} } or H = V; otherwise, the ordered basis for H can be extended by two more elements to an ordered basis of a J-invariant subspace as in the Proof of Lemma 5.13. Note that this Lemma then applies to U and can be repeated to get another subspace containing U.

Lemma 5.18. Given a vector space V_{1} with CSO J_{1} and an element v ∈ V_{1}, another vector space V_{2} with CSO J_{2}, and a real linear map A : V_{1} → V_{2}, the following are equivalent.

(1) A(J_{1}(v)) ∈ span{A(v), J_{2}(A(v))}.
(2) A maps the subspace span{v, J_{1}(v)} ⊆ V_{1} to the subspace
span{A(v), J_{2}(A(v))} ⊆ V_{2}.

Further, if A and v satisfy (1) and A(J_{1}(v)) ≠ 0_{V_{2}}, then A and J_{1}(v) satisfy (1):

A(J_{1}(J_{1}(v))) ∈ span{A(J_{1}(v)), J_{2}(A(J_{1}(v)))}.

Proof. (2) ⟺ (1) is straightforward. If A(J_{1}(v)) = α_{1} ·A(v)+α_{2} ·J_{2}(A(v)) for some (α_{1}, α_{2}) ≠ (0, 0), then there is this linear combination.

\frac{−α_{1}}{α_{1}^{2} + α_{2}^{2}} · A(J_{1}(v)) + \frac{α_{2}}{α_{1}^{2} + α_{2}^{2}} · J_{2}(A(J_{1}(v)))

= \frac{−α_{1}}{α_{1}^{2} + α_{2}^{2}} · (α_{1} · A(v) + α_{2} · J_{2}(A(v)))

+\frac{α_{2}}{α_{1}^{2} + α_{2}^{2}} · J_{2}(α_{1} · A(v) + α_{2} · J_{2}(A(v)))

= −A(v) = A(J_{1}(J_{1}(v))).

The above notion for a real linear map A is slightly stronger than the statement that A maps the J_{1}-complex line span{v, J_{1}(v)} into some J_{2}-complex line; if A(v) = 0_{V_{2}}, condition (2) implies A maps span{v, J_{1}(v)} to the zero subspace.

Exercise 5.19. Given V with CSO J, if V ≠ {0_{V} }, then the ordered list (Id_{V} , J) is linearly independent in End(V).

Exercise 5.20. Given V with two CSOs J_{1} and J_{2}, if J_{2} ≠ ±J_{1}, then the ordered list (Id_{V} , J_{1}, J_{2}, J_{1} ∘ J_{2}) is linearly independent in End(V).

Hint. The hypothesis implies V ≠ {0_{V} }, so Exercise 5.19 applies and (Id_{V} , J_{1}) is a linearly independent list. The next step is to show that (Id_{V} , J_{1}, J_{2}) is a linearly independent list. If there are real scalars such that α_{1}·Id_{V} +α_{2}·J_{1}+α_{3}·J_{2} = 0_{End(V)}, then:

(α_{1} · Id_{V} + α_{2} · J_{1}) ∘ (α_{1} · Id_{V} + α_{2} · J_{1}) = (−α_{3} · J_{2}) ∘ (−α_{3} · J_{2})
(α_{1}^{2} − α_{2}^{2}) · Id_{V} + 2α_{1}α_{2} · J_{1} = −α_{3}^{2} · Id_{V}
⟹ α_{1}α_{2} = 0.

If α_{2} = 0 then by the linear independence of (Id_{V} , J_{2}) from Exercise 5.19, α_{1} = α_{2} = α_{3} = 0. If α_{1} = 0 then (α_{2} · J_{1}) ∘ (α_{2} · J_{1}) = (−α_{3} · J_{2}) ∘ (−α_{3} · J_{2}) ⟹ −α_{2}^{2} = −α_{3}^{2} ⟹ α_{2} · (J_{1} ± J_{2}) = 0_{End(V)}, and the hypothesis J_{2} ≠ ±J_{1} implies α_{1} = α_{2} = α_{3} = 0.

The claimed independence of the list (Id_{V} , J_{1}, J_{2}, J_{1} ∘ J_{2}) then follows from applying Lemma 5.11 to the list (Id_{V} , J_{1}, J_{2}) and the CSO Hom(J_{2}, Id_{V}) on End(V) from Example 5.4.

Remark 5.21. The results in this Section give some details omitted from [C_{1}] §2.

5.2. Complex linear and antilinear maps

Definition 5.22. For U = (U, J_{U}), V = (V, J_{V}), a (real linear) map A ∈ Hom(U, V) is \underline{c-linear}, means: A ∘ J_{U} = J_{V} ∘ A. A map A ∈ Hom(U, V) is \underline{a-linear} means: A ∘ J_{U} = −J_{V} ∘ A.

Because some vector spaces can admit several complex structures, it will sometimes be more clear to specifically refer to A : U → V as c-linear (or a-linear) with respect to the pair (J_{U}, J_{V}).

Lemma 5.23. If A : U → V is c-linear (or a-linear) and invertible, then A^{−1} is also c-linear (or a-linear). The composite of two c-linear maps (or two a-linear maps) is c-linear.

Lemma 5.24. Given V = (V, J_{V}) and V′ = (V′, J′_{V}), any map A : U → U′, and a c-linear map B : V → V′, the maps [A ⊗ B] : U ⊗ V → U′ ⊗ V′ and [B ⊗ A] : V ⊗ U → V′ ⊗ U′ are c-linear with respect to the induced CSOs from Example 5.3.

Lemma 5.25. Given V = (V, J_{V}) and V′ = (V′, J′_{V}), any map A : U′ → U, and a c-linear map B : V → V′, the maps Hom(A, B) : Hom(U, V) → Hom(U′, V′) and Hom(B, A) : Hom(V′, U′) → Hom(V, U) are c-linear with respect to the induced CSOs from Example 5.4.

Lemma 5.26. Given V and a CSO J ∈ End(V), for any invertible A : U → V, the composite A^{−1}∘J ∘A ∈ End(U) is a CSO. A is c-linear with respect to A^{−1}∘J ∘A and J. If B : U → V is another invertible map and A^{−1} ∘ J ∘ A = B^{−1} ∘ J ∘ B, then A ∘ B^{−1} is a c-linear endomorphism of (V, J).

The CSO A^{−1} ∘ J ∘ A is the \underline{pullback} of J.

Given a direct sum V = V_{1} ⊕ V_{2}, recall from Definition 1.84 that a CSO J ∈ End(V) respects the direct sum if P_{I} ∘ J ∘ Q_{i} = 0_{Hom(V_{i},V_{I})} for i ≠ I, or equivalently, Q_{i} ∘ P_{i} ∘ J = J ∘ Q_{i} ∘ P_{i} for i = 1 or 2, so the map Q_{i} ∘ P_{i} is c-linear.

Lemma 5.27. Given V = V_{1} ⊕ V_{2} and CSOs J_{V_{i}} ∈ End(V_{i}) for i = 1, 2, then Q_{1} ∘ J_{V_{1}} ∘ P_{1} + Q_{2} ∘ J_{V_{2}} ∘ P_{2} ∈ End(V) is a CSO, and it respects the direct sum.

Lemma 5.28. Given V = V_{1} ⊕ V_{2} and a CSO J ∈ End(V) that respects the direct sum, each induced map P_{i} ∘ J ∘ Q_{i} on V_{i} is a CSO.

Proof. The claim is easily checked. This Lemma is compatible with the previous one: given J, it is recovered by re-combining the induced maps as in Lemma 5.27:

Q_{1} ∘ (P_{1} ∘ J ∘ Q_{1}) ∘ P_{1} + Q_{2} ∘ (P_{2} ∘ J ∘ Q_{2}) ∘ P_{2} = J.

Conversely, given J_{V_{1}}, J_{V_{2}} as in Lemma 5.27, they agree with CSOs induced by Q_{1} ∘ J_{V_{1}} ∘ P_{1} + Q_{2} ∘ J_{V_{2}} ∘ P_{2}:

P_{i} ∘ (Q_{1} ∘ J_{V_{1}} ∘ P_{1} + Q_{2} ∘ J_{V_{2}} ∘ P_{2}) ∘ Q_{i} = J_{V_{i}}.

Lemma 5.29. For V = V_{1} ⊕ V_{2} and V′ = V′_{1} ⊕ V′_{2} and invertible maps A : V_{2} → V_{1}, A′ : V′_{2} → V′_{1}, let J, J′ be CSOs on V and V′ constructed as in Example 5.8. Then, for H : V → V′, the following are equivalent.

(1) H is c-linear with respect to J and J′.

(2) A′ ∘ P′_{2} ∘ H ∘ Q_{2} = P′_{1} ∘ H ∘ Q_{1} ∘ A and P′_{1} ∘ H ∘ Q_{2} = −A′ ∘ P′_{2} ∘ H ∘ Q_{1} ∘ A.

Hint. To show (2) ⟹ (1), expand

H ∘ J = (Q′_{1} ∘ P′_{1} + Q′_{2} ∘ P′_{2}) ∘ H ∘ (Q_{2} ∘ A^{−1} ∘ P_{1} − Q_{1} ∘ A ∘ P_{2})

and similarly J′ ∘ H.

For (1) ⟹ (2), apply Hom(Q_{1}, P′_{2}) to both sides of H ∘ J = J′ ∘ H to get one of the equations, and apply Hom(Q_{2}, P′_{2}) to get the other.

Remark 5.30. The preceding Lemma displays an algebraic pattern analogous to the Cauchy-Riemann equations.

194 5. COMPLEX STRUCTURES

Exercise 5.31. Given (V_{1}, J_{1}) and (V_{2}, J_{2}) and V = V_{1}⊕V_{2}, a map A : V_{1} → V_{2} is a-linear if and only if

Q_{1} ∘ J_{1} ∘ P_{1} + Q_{2} ∘ A ∘ P_{1} + Q_{2} ∘ J_{2} ∘ P_{2}

is a CSO on V. This CSO respects the direct sum if and only if A = 0_{Hom(V_{1},V_{2})}, in which case it is the CSO constructed in Lemma 5.27.

Exercise 5.32. For V, V_{1}, V_{2}, A as in the previous Exercise, the CSO J_{A} = Q_{1} ∘ J_{1} ∘ P_{1} + Q_{2} ∘ A ∘ P_{1} + Q_{2} ∘ J_{2} ∘ P_{2} is similar to the direct sum CSO J_{0} = Q_{1} ∘ J_{1} ∘ P_{1} + Q_{2} ∘ J_{2} ∘ P_{2}, in the sense that J_{A} = G^{−1} ∘ J_{0} ∘ G for some invertible G ∈ End(V), and G can be chosen so that P_{2} ∘ G ∘ Q_{2} = Id_{V_{2}}.

Hint. Let G = Id_{V} + \frac{1}{2} · Q_{2} ∘ A ∘ J_{1} ∘ P_{1}, then check G ∘ J_{A} = J_{0} ∘ G, or use G^{−1} = Id_{V} − \frac{1}{2} · Q_{2} ∘ A ∘ J_{1} ∘ P_{1}.

Exercise 5.33. Given V and A ∈ End(V), if V is finite-dimensional and there is some CSO J on V such that A is a-linear with respect to J, then T r_{V} (A) = 0.

Exercise 5.34. Given V = (V, J_{V}), the canonical maps l_{1} : ℝ ⊗ V → V and l_{2} : V ⊗ ℝ → V from Example 1.28 are c-linear.

Hint. The CSO on ℝ ⊗ V is as in Example 5.3, and the claim for l_{1} follows from Lemma 1.38. The claim for l_{2} : V ⊗ ℝ → V is analogous.

Exercise 5.35. Given W = (W, J), the map m : W → Hom(𝕂, W) from Definition 1.20 is c-linear.

Hint. The CSO on Hom(𝕂, W) is as in Example 5.4, and the claim follows from Lemma 1.21.

Exercise 5.36. Given U, V, W, with U = (U, J), the canonical map (Definition 1.7) t_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W), Hom(U, W)) is c-linear with respect to the induced CSOs as in Example 5.4.

Exercise 5.37. Given U, V, W, with V = (V, J), t_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W), Hom(U, W)) is c-linear with respect to the induced CSOs.

Exercise 5.38. Given U, V, W, with W = (W, J), and any A ∈ Hom(U, V), the map

t_{UV}^{W} (A) ∈ Hom(Hom(V, W), Hom(U, W))

is c-linear Hom(V, W) → Hom(U, W).

Hint. This claim, Exercise 5.36, and Exercise 5.37 all follow from Lemma 1.8.

Exercise 5.39. Given V and W, with V = (V, J_{V}), the canonical map (Definition 1.13) d_{V W} : V → Hom(Hom(V, W), W) is c-linear with respect to the induced CSO as in Example 5.4.

Exercise 5.40. Given V and W, with W = (W, J_{W}), and any v ∈ V, the map d_{V W} (v) ∈ Hom(Hom(V, W), W) is c-linear Hom(V, W) → W.

Hint. This claim and Exercise 5.39 both follow from Lemma 1.14.

Exercise 5.41. Given U, V, W, with U = (U, J), the canonical map (Definition 1.52) e_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W) ⊗ U, W) is c-linear with respect to the induced CSOs.

Exercise 5.42. Given U, V, W, with V = (V, J), e_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W) ⊗ U, W) is c-linear with respect to the induced CSOs.

Exercise 5.43. Given U, V, W, with W = (W, J), and any A ∈ Hom(U, V), the map e_{UV}^{W} (A) ∈ Hom(Hom(V, W) ⊗ U, W) is c-linear Hom(V, W) ⊗ U → W.

Hint. This claim, Exercise 5.41, and Exercise 5.42 all follow from Lemma 1.53.

Recall from Definition 2.70 that Hom(Hom(V, W) ⊗ V, W) contains a distinguished element Ev_{V W} : A ⊗ v ↦ A(v).

Exercise 5.44. Given V and W, with W = (W, J_{W}), the canonical evaluation Ev_{V W} : Hom(V, W) ⊗ V → W is c-linear.

Hint. This claim can be checked directly; it also follows from Lemma 2.72, or the formula Ev_{V W} = e_{V V}^{W} (Id_{V}) from Equation (2.12) and Exercise 5.43.

Exercise 5.45. Given U, V, W, with W = (W, J_{W}), the map

T_{U,V;W} : Hom(U, Hom(V, W)) → Hom(V, Hom(U, W))

from Definition 4.2 is c-linear.

Hint. The claim follows from Lemma 4.6.

Exercise 5.46. Given U, V, W, with U = (U, J_{U}), the map

T_{U,V;W} : Hom(U, Hom(V, W)) → Hom(V, Hom(U, W)),

and its inverse from Equation (4.2) in Lemma 4.4,

T_{V,U;W} : Hom(V, Hom(U, W)) → Hom(U, Hom(V, W)),

are c-linear.

Hint. Both claims follow from Lemma 4.6. Alternatively, the c-linearity of the composite formula from Definition 4.2,

T_{U,V;W} = Hom(d_{V W} , Id_{Hom(U,W)}) ∘ t_{U,Hom(V,W)}^{W},

and its inverse could be shown to follow from the c-linearity of the t and d maps, as in Exercise 5.36, Exercise 5.37, and Exercise 5.39.

Proposition 5.47. Given V, consider three elements A, J_{1}, J_{2} ∈ End(V). The following two statements are equivalent.

(1) (J_{1} + J_{2}) ∘ A = J_{1} − J_{2}.

(2) J_{2} ∘ (Id_{V} + A) = J_{1} ∘ (Id_{V} − A).

The following two statements are also equivalent to each other.

(1′) A ∘ (J_{1} + J_{2}) = J_{1} − J_{2}.

(2′) (Id_{V} + A) ∘ J_{2} = (Id_{V} − A) ∘ J_{1}.

If A, J_{1}, J_{2} satisfy either condition (1) or (1′), then any two of the following imply the remaining third.

(3) J_{1} is invertible.

(4) J_{1} + J_{2} is invertible.
(5) Id_{V} + A is invertible.

If A, J_{1}, J_{2} satisfy (1) or (1′), and also (5), then any two of the following imply the remaining third.

(6) A ∘ J_{1} = −J_{1} ∘ A.

(7) J_{1} is a CSO.

(8) J_{2} is a CSO.

Proof. (1) ⟺ (2) by an elementary algebraic manipulation. The (1′) ⟺ (2′) and subsequent implications are analogous and left as an exercise. For (3), (4), (5), use (1) or (2) to establish

(J_{1} + J_{2}) ∘ (Id_{V} + A) = 2 · J_{1},

and the claim follows.

For (6), (7), (8), use (2) to establish

(J_{1} + J_{2}) ∘ (A ∘ J_{1} + J_{1} ∘ A) (5.4)

= J_{1} ∘ A ∘ J_{1} + J_{2} ∘ A ∘ J_{1} + J_{1} ∘ J_{1} ∘ A + J_{2} ∘ J_{1} ∘ A
= J_{1} ∘ A ∘ J_{1} + J_{2} ∘ (Id_{V} + A) ∘ J_{1} + J_{1} ∘ J_{1} ∘ A − J_{2} ∘ J_{1} ∘ (Id_{V} − A)
= J_{1} ∘ A ∘ J_{1} + J_{1} ∘ (Id_{V} − A) ∘ J_{1} + J_{1} ∘ J_{1} ∘ A − J_{2} ∘ J_{2} ∘ (Id_{V} + A)
= (J_{1} ∘ J_{1} − J_{2} ∘ J_{2}) ∘ (Id_{V} + A).

Given (6), (8), the equation (5.4) becomes 0_{End(V)} = (J_{1} ∘ J_{1} + Id_{V}) ∘ (Id_{V} + A), and (7) follows from (5). Similarly, given (6), (7), (5.4) becomes 0_{End(V)} = −(Id_{V} + J_{2} ∘ J_{2}) ∘ (Id_{V} + A), and (8) follows from (5). For (7), (8), note that (7) implies (3) and then (5) implies (4), so by (7) and (8), (5.4) becomes LHS = 0_{End(V)}, and (6) follows from (4).

Given V with a CSO J_{1}, Proposition 5.47 establishes a bijective correspondence between the set of CSOs J_{2} on V with J_{1} +J_{2} invertible and the set of A ∈ End(V) with A a-linear (with respect to J_{1}) and Id_{V} +A invertible, as follows. Since J_{1} is a CSO, (3) and (7) hold. For any a-linear A ∈ End(V) with Id_{V} +A invertible, (5) and (6) hold. If we define J_{2} by the similarity relation J_{2} = (Id_{V} +A)∘J_{1} ∘(Id_{V} +A)^{−1}, then J_{2} = J_{1} ∘ (Id_{V} − A) ∘ (Id_{V} + A)^{−1}, so (2) holds, (1), (4), and (8) follow as consequences, and A satisfies A = (J_{1} + J_{2})^{−1} ∘ (J_{1} − J_{2}). Conversely, for any CSO J_{2} with J_{1} + J_{2} invertible, (4) and (8) hold. If we define

(5.5)  A = (J_{1} + J_{2})^{−1} ∘ (J_{1} − J_{2}),

then (1) holds, (2), (5), and (6) follow as consequences, and J_{2} satisfies J_{2} = J_{1} ∘ (Id_{V} − A) ∘ (Id_{V} + A)^{−1} = (Id_{V} + A) ∘ J_{1} ∘ (Id_{V} + A)^{−1}.

Exercise 5.48. Given V and any two CSOs J_{1}, J_{2}, the map J_{1} + J_{2} is c-linear with respect to J_{1} and J_{2}, and the maps ±(J_{1} − J_{2}) are a-linear with respect to J_{1} and J_{2}.

Big Exercise 5.49. Given V_{1}, V_{2} with CSOs J_{1}, J_{2}, and a real linear map A : V_{1} → V_{2}, if the image subspace A(V_{1}) admits a linearly independent list of 3 or more elements of V_{2}, then the following are equivalent.

(1) For each v ∈ V_{1}, A(J_{1}(v)) ∈ span{A(v), J_{2}(A(v))}.

(2) For each v ∈ V_{1}, A maps the subspace span{v, J_{1}(v)} ⊆ V_{1} to the subspace

span{A(v), J_{2}(A(v))} ⊆ V_{2}.

(3) For each v ∈ V_{1}, either A(J_{1}(v)) = J_{2}(A(v)) or A(J_{1}(v)) = −J_{2}(A(v)).

(4) A ∘ J_{1} = J_{2} ∘ A or A ∘ J_{1} = −J_{2} ∘ A.

Hint. The idea is that A takes J_{1}-complex lines to J_{2}-complex lines, and that this is equivalent to A being c-linear or a-linear. See also [C_{1}] for other properties of A equivalent to (4).

5.3. Commuting Complex Structure Operators

5.3.1. Two Commuting Complex Structure Operators.

Lemma 5.50. Given V and two CSOs J_{1}, J_{2}, the following are equivalent.

(1) J_{1} and J_{2} commute (i.e., J_{1} ∘ J_{2} = J_{2} ∘ J_{1}).

(2) The composite J_{1} ∘ J_{2} is an involution.

(3) The composite −J_{1} ∘ J_{2} is an involution.

Example 5.51. Given V with commuting CSOs J_{1}, J_{2}, Lemma 1.114 applies to the involution −J_{1} ∘ J_{2} as in Lemma 5.50: there is a direct sum V = V_{c} ⊕ V_{a} produced by −J_{1} ∘ J_{2}, where

V_{c} = {v ∈ V : (−J_{1} ∘ J_{2})(v) = v} = {v ∈ V : J_{2}(v) = J_{1}(v)}
V_{a} = {v ∈ V : (−J_{1} ∘ J_{2})(v) = −v} = {v ∈ V : J_{2}(v) = −J_{1}(v)},

with projection operators

P_{c} = \frac{1}{2} · (Id_{V} − J_{1} ∘ J_{2}) : V ↠ V_{c}, P_{a} = \frac{1}{2} · (Id_{V} + J_{1} ∘ J_{2}) : V ↠ V_{a}.

As remarked in the Proof of Lemma 1.114, the same formulas are also used for Q_{c} ∘ P_{c}, Q_{a} ∘ P_{a} ∈ End(V), where Q_{c} and Q_{a} are the corresponding subspace inclusions.

Note that applying Lemma 1.114 to the involution J_{1} ∘J_{2} would give the direct sum in the other order, V = V_{a} ⊕ V_{c}.

Lemma 5.52. For V, J_{1}, J_{2} as in Lemma 5.50, and another space V′ with commuting CSOs J′_{1}, J′_{2}, a map H : V → V′ respects the direct sums V_{c} ⊕ V_{a} and V′_{c} ⊕ V′_{a} if and only if H ∘ J_{1} ∘ J_{2} = J′_{1} ∘ J′_{2} ∘ H.

Proof. This is an example of Lemma 1.120.

Example 5.53. Lemma 5.52 applies to V′ = V, J′_{1} = J_{1}, J′_{2} = J_{2}, and either H = J_{1} or H = J_{2}, each of which induces a CSO on V_{c} and on V_{a} by Lemma 5.28. The subspace V_{c} has a canonical CSO, induced by either J_{1} or J_{2}:

P_{c} ∘ J_{1} ∘ Q_{c} = P_{c} ∘ J_{2} ∘ Q_{c} ∈ End(V_{c}).

The maps P_{c} : V ↠ V_{c} and Q_{c} : V_{c} ↪ V are c-linear with respect to either CSO on V. The induced CSOs on the subspace V_{a} are opposite, and generally distinct:

P_{a} ∘ J_{1} ∘ Q_{a} = −P_{a} ∘ J_{2} ∘ Q_{a} ∈ End(V_{a}).

Lemma 5.54. For V, J_{1}, J_{2} as in Lemma 5.50, U = (U, J_{U}), and a map H : U → V, if H is c-linear with respect to both (J_{U}, J_{1}) and (J_{U}, J_{2}), then the image of H is contained in V_{c}.

Proof. This can be checked by showing P_{a} ∘ H = 0_{Hom(U,V_{a})}.

Lemma 5.55. For V, J_{1}, J_{2} as in Lemma 5.50, U = (U, J_{U}), and a map H : V → U, if H is c-linear with respect to either (J_{1}, J_{U}) or (J_{2}, J_{U}), then H ∘ Q_{c} : V_{c} → U is c-linear.

Proof. By the construction of the CSO on V_{c} from Example 5.53, the LHS quantities in the following two equations are equal to each other:

(H ∘ Q_{c}) ∘ (P_{c} ∘ J_{1} ∘ Q_{c}) = H ∘ J_{1} ∘ Q_{c},
(H ∘ Q_{c}) ∘ (P_{c} ∘ J_{2} ∘ Q_{c}) = H ∘ J_{2} ∘ Q_{c}.

The equalities hold because J_{1} and J_{2} commute with Q_{c} ∘ P_{c}. On the RHS, either H∘J_{1} or H∘J_{2} is equal to J_{U} ∘H by hypothesis, so the claim (H∘Q_{c})∘(P_{c}∘J_{1}∘Q_{c}) = J_{U} ∘ (H ∘ Q_{c}) follows.

Lemma 5.56. For V with commuting CSOs J_{1}, J_{2}, and V′ with commuting CSOs J′_{1}, J′_{2} as in Lemma 5.52, if H : V → V′ is c-linear with respect to any of the pairs (J_{1}, J′_{1}) or (J_{1}, J′_{2}) or (J_{2}, J′_{1}) or (J_{2}, J′_{2}), then P′_{c} ∘ H ∘ Q_{c} : V_{c} → V′_{c} is c-linear with respect to the canonical CSOs.

Proof. The CSOs on V_{c} and V′_{c} are as in Example 5.53.

Lemma 5.57. Given V with commuting CSOs J_{V}^{1} , J_{V}^{2} , and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, if H : U → V satisfies both H ∘ J_{U}^{1} = J_{V}^{1} ∘ H and H ∘ J_{U}^{2} = J_{V}^{2} ∘ H, then H : U_{c} ⊕ U_{a} → V_{c} ⊕ V_{a} respects the direct sums and the induced map P_{c}^{V} ∘ H ∘ Q_{c}^{U} : U_{c} → V_{c} is c-linear with respect to the induced CSOs. If, also, H is invertible, then for i = c, a, the induced map P_{i}^{V} ∘ H ∘ Q_{i}^{U} : U_{i} → V_{i} is invertible.

Proof. This follows from Lemma 5.52, Lemma 5.56, and Lemma 1.85.

We remark that the direct summands in Lemma 5.57 are all subspaces; if u = Q_{c}^{U}(u) ∈ U_{c} ⊆ U, then H(u) = H(Q_{c}^{U}(u)) ∈ V_{c} ⊆ V, so H(u) is in the fixed point set of the idempotent P_{c}^{V} : V → V, as follows:

u ∈ U_{c} ⟹ H(u) = H(Q_{c}^{U}(u)) = H(Q_{c}^{U}(P_{c}^{U}(Q_{c}^{U}(u)))) = Q_{c}^{V} (P_{c}^{V} (H(Q_{c}^{U}(u))))
= (P_{c}^{V} ∘ H ∘ Q_{c}^{U})(u) = P_{c}^{V} (H(u)) ∈ V_{c} ⊆ V.

The induced map P_{c}^{V} ∘ H ∘ Q_{c}^{U} : U_{c} → V_{c} is just the restriction of H : U → V to the subspace U_{c}, with image contained in the subspace V_{c} of the target.

Lemma 5.58. Given V with commuting CSOs J_{V}^{1} , J_{V}^{2} , and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, if H : U → V satisfies both H ∘ J_{U}^{1} = −J_{V}^{1} ∘ H and H ∘ J_{U}^{2} = −J_{V}^{2} ∘ H, then H : U_{c} ⊕ U_{a} → V_{c} ⊕ V_{a} respects the direct sums and the induced map P_{c}^{V} ∘ H ∘ Q_{c}^{U} : U_{c} → V_{c} is a-linear with respect to the induced CSOs.

Proof. This is straightforward to check directly, or follows from Lemma 5.57 with J_{V}^{1} , J_{V}^{2} replaced by the opposite CSOs −J_{V}^{1} , −J_{V}^{2} .

The following Theorem weakens the hypotheses of Lemma 5.57.

Theorem 5.59. Given V with commuting CSOs J_{V}^{1} , J_{V}^{2} , and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, if H : U → V is c-linear with respect to (J_{U}^{2}, J_{V}^{2} ), then the kernel of the composite P_{c}^{V} ∘ H ∘ Q_{a}^{U} : U_{a} → V_{c} is equal to the set {u ∈ U_{a} : (H ∘ J_{U}^{1} ∘ Q_{a}^{U})(u) = (J_{V}^{1} ∘ H ∘ Q_{a}^{U})(u)}.

Proof. Composing with Q_{c}^{V} does not change the kernel, so using the equalities J_{U}^{1} ∘ Q_{a}^{U} = −J_{U}^{2} ∘ Q_{a}^{U} and J_{V}^{2} ∘ H = H ∘ J_{U}^{2},

Q_{c}^{V} ∘ P_{c}^{V} ∘ H ∘ Q_{a}^{U} = \frac{1}{2} · (Id_{V} − J_{V}^{1} ∘ J_{V}^{2} ) ∘ H ∘ Q_{a}

= \frac{1}{2} · (H − J_{V}^{1} ∘ H ∘ J_{U}^{2}) ∘ Q_{a}^{U} = \frac{1}{2} · (H + J_{V}^{1} ∘ H ∘ J_{U}^{1}) ∘ Q_{a}^{U},

and the composite with the invertible map 2 · J_{V}^{1} has the same kernel:

2 · J_{V}^{1} ∘ Q_{c}^{V} ∘ P_{c}^{V} ∘ H ∘ Q_{a}^{U} = J_{V}^{1} ∘ H ∘ Q_{a}^{U} − H ∘ J_{U}^{1} ∘ Q_{a}^{U}.

Example 5.60. For a vector space V with a CSO J and involution N that commute as in Example 5.6, the CSO N ∘J commutes with J and N. The involution N produces a direct sum V = V_{1} ⊕ V_{2} with projections (P_{1}, P_{2}) and inclusions (Q_{1}, Q_{2}) as in Lemma 1.114, and Lemma 1.120 and Lemma 5.28 apply to both J and N ∘J: they respect the direct sum V_{1}⊕V_{2}, and the induced maps P_{1}∘J ∘Q_{1} and P_{1} ∘ N ∘ J ∘ Q_{1} are commuting CSOs on V_{1}, and similarly for V_{2}. More specifically, the CSOs on V_{1} are equal: P_{1} ∘ J ∘ Q_{1} = P_{1} ∘ N ∘ J ∘ Q_{1}, while the CSOs P_{2} ∘ J ∘ Q_{2} and P_{2} ∘ N ∘ J ∘ Q_{2} on V_{2} are opposite.

Example 5.61. Given U with commuting CSOs J_{1}, J_{2}, if H ∈ End(U) is an involution such that H ∘ J_{1} ∘ J_{2} = J_{1} ∘ J_{2} ∘ H, then U admits two direct sums: U = U_{1} ⊕ U_{2} produced by H as in Lemma 1.114, and U = U_{c} ⊕ U_{a} produced by −J_{1} ∘ J_{2}. The composite (−J_{1} ∘ J_{2}) ∘ H is a third involution, which produces a direct sum U = U_{5} ⊕ U_{6}, as denoted in Theorem 1.124. Both H and −J_{1} ∘ J_{2} ∘ H respect the direct sum U_{c} ⊕ U_{a} as in Lemma 1.122 and Lemma 5.52; they both induce involutions U_{c} → U_{c} and U_{a} → U_{a}, with the involutions being equal on U_{c} as in (1.13), and opposite on U_{a} as in (1.14), producing a direct sum U_{c} = U′_{c} ⊕ U′′_{c}, and an unordered pair of subspaces of U_{a}:

(5.6)  U′_{c} = {u ∈ U : u = −J_{1}(J_{2}(u)) = H(u)},
U′′_{c} = {u ∈ U : u = −J_{1}(J_{2}(u)) = −H(u)},
U′_{a} = {u ∈ U : u = J_{1}(J_{2}(u)) = H(u)},
U′′_{a} = {u ∈ U : u = J_{1}(J_{2}(u)) = −H(u)}.

Similarly, both −J_{1} ∘ J_{2} and −J_{1} ∘ J_{2} ∘ H respect the direct sum U_{1} ⊕ U_{2}, and induce involutions on U_{1} and U_{2} that distinguish the same four subspaces:

U′_{1} = {u ∈ U : u = H(u) = −J_{1}(J_{2}(u))},
U′′_{1} = {u ∈ U : u = H(u) = J_{1}(J_{2}(u))},
U′_{2} = {u ∈ U : u = −H(u) = −J_{1}(J_{2}(u))},
U′′_{2} = {u ∈ U : u = −H(u) = J_{1}(J_{2}(u))}.

This configuration of subspaces gives an example of the results of Theorem 1.124, U′_{1} = U′_{c} = U_{1} ∩ U_{c} = U_{1} ∩ U_{c} ∩ U_{5}, U′′_{c} = U′_{2} = U_{2} ∩ U_{6}, U′′_{1} = U′_{a} = U_{a} ∩ U_{6}, etc.

Lemma 5.62. Given U with commuting CSOs J_{1}, J_{2}, if H ∈ End(U) is an involution such that H ∘ J_{1} = J_{1} ∘ H and H ∘ J_{2} = J_{2} ∘ H, then H respects the direct sum U_{c} ⊕ U_{a} → U_{c} ⊕ U_{a}, and induces involutions on U_{c} and U_{a}. The induced involution P_{c} ∘ H ∘ Q_{c} on U_{c} is c-linear, and its fixed point set U′_{c} has a canonical CSO.

Proof. It follows from the c-linearity of H that H ∘ J_{1} ∘ J_{2} = J_{1} ∘ J_{2} ∘ H. So, this is a special case of both Lemma 5.57 and Example 5.61: the induced involution on U_{c} is c-linear with respect to the CSO on U_{c} from Example 5.53, and produces a direct sum U_{c} = U′_{c}⊕U′′_{c} as in (5.6). The subspace U′_{c} = {u ∈ U : u = −J_{1}(J_{2}(u)) = H(u)} has a canonical CSO, from Example 5.60.

Lemma 5.63. Given U with commuting CSOs J_{1}, J_{2}, if H ∈ End(U) is an involution such that H ∘ J_{1} = J_{2} ∘ H, then H respects the direct sum U_{c} ⊕ U_{a} → U_{c} ⊕ U_{a}, and induces involutions on U_{c} and U_{a}. The induced involution P_{c} ∘ H ∘ Q_{c} on U_{c} is c-linear, and its fixed point set U′_{c} has a canonical CSO.

Proof. It follows from the involution property that H ∘ J_{1} = J_{2} ∘ H ⟹ J_{1} ∘ H = H ∘ J_{2} ⟹ H ∘ J_{1} ∘ J_{2} = J_{1} ∘ J_{2} ∘ H. So, this is similar to Lemma 5.62, and also a special case of both Lemma 5.57 and Example 5.61: the induced involution on U_{c} is c-linear and produces a direct sum U_{c} = U′_{c} ⊕ U′′_{c}. The subspace U′_{c} = {u ∈ U : u = −J_{1}(J_{2}(u)) = H(u)} has a canonical CSO, from Example 5.60.

Example 5.64. Given U and V, suppose there are commuting CSOs J_{1}, J_{2} on V. Then the CSOs [J_{1} ⊗ Id_{U}] and [J_{2} ⊗ Id_{U}] on V ⊗ U, from Example 5.3, also commute. As in Example 5.51, this gives a direct sum V ⊗U = (V ⊗U)_{c} ⊕(V ⊗U)_{a} with projections

\frac{1}{2} · (Id_{V ⊗U} ± [J_{1} ⊗ Id_{U}] ∘ [J_{2} ⊗ Id_{U}]).

V ⊗ U also admits a direct sum (V_{c} ⊗ U) ⊕ (V_{a} ⊗ U) as in Example 1.77, with projections

[(\frac{1}{2} · (Id_{V} ± J_{1} ∘ J_{2})) ⊗ Id_{U}] = [P_{i} ⊗ Id_{U}],

for P_{i} : V ↠ V_{i}, i = c, a, as in Example 5.51. This is a special case of Example 1.135; the pairs of projections are identical (using the linearity of j and Lemma 1.36), so the direct sums are the same and we have the equalities of subspaces (V ⊗ U)_{c} = V_{c} ⊗ U and (V ⊗ U)_{a} = V_{a} ⊗ U, with inclusions [Q_{i} ⊗ Id_{U}]. Similarly, (U ⊗ V)_{c} = U ⊗ V_{c} and (U ⊗ V)_{a} = U ⊗ V_{a}.

Example 5.65. Given U = (U, J_{U}) and V = (V, J_{V}), the two CSOs [Id_{U} ⊗J_{V}], [J_{U} ⊗ Id_{V}] ∈ End(U ⊗ V) commute, so this is a special case of Example 5.51. The direct sum so produced is denoted

U ⊗ V = (U ⊗_{c} V) ⊕ (U ⊗_{a} V).

As in Example 5.53, the subspace

U ⊗_{c} V = {w ∈ U ⊗ V : [Id_{U} ⊗ J_{V}](w) = [J_{U} ⊗ Id_{V}](w)}

has a canonical CSO, induced by either of the CSOs, so we may denote the space U ⊗_{c} V. The CSOs on U ⊗ V induce opposite CSOs on the subspace

U ⊗_{a} V = {w ∈ U ⊗ V : [Id_{U} ⊗ J_{V}](w) = −[J_{U} ⊗ Id_{V}](w)}.

Example 5.66. For c-linear maps A : U → U′ and B : V → V′, the map

[A ⊗ B] : U ⊗ V → U′ ⊗ V′

satisfies the hypotheses of Lemma 5.57, and respects the direct sums from Example 5.65. The induced map

P′_{c} ∘ [A ⊗ B] ∘ Q_{c} : U ⊗_{c} V → U′ ⊗_{c} V′

is c-linear.

Notation 5.67. For c-linear maps A : U → U′ and B : V → V′ as in Example 5.66, the bracket notation from Notation 1.35 is adapted in the following abbreviation:

[A ⊗_{c} B] = P′_{c} ∘ [A ⊗ B] ∘ Q_{c} : U ⊗_{c} V → U′ ⊗_{c} V′.

As remarked after Lemma 5.57, this is exactly the restriction of the map [A ⊗ B] to the U⊗_{c} V subspace of the domain, with image contained in the U′ ⊗_{c} V′ subspace of the target. The role of the j map in this construction is considered in more detail by Example 5.118.

Exercise 5.68. For a-linear maps A : U → U′ and B : V → V′, the map

[A ⊗ B] : U ⊗ V → U′ ⊗ V′

respects the direct sums. The induced map

P′_{c} ∘ [A ⊗ B] ∘ Q_{c} : U ⊗_{c} V → U′ ⊗_{c} V′

is a-linear.

Example 5.69. Given U = (U, J_{U}) and V = (V, J_{V}), the CSO Hom(Id_{U}, J_{V}) commutes with Hom(J_{U}, Id_{V}) ∈ End(Hom(U, V)), so this is a special case of Example 5.51. The direct sum so produced is denoted

(5.7)  Hom(U, V) = Hom_{c}(U, V) ⊕ Hom_{a}(U, V).

The projection P_{c} : Hom(U, V) ↠ Hom_{c}(U, V) is defined by

\frac{1}{2} ·(Id_{Hom(U,V)} −Hom(Id_{U}, J_{V})∘Hom(J_{U}, Id_{V})) = \frac{1}{2} ·(Id_{Hom(U,V)} −Hom(J_{U}, J_{V})).

As in Example 5.53, the subspace

Hom_{c}(U, V) = {A ∈ Hom(U, V) : Hom(Id_{U}, J_{V})(A) = Hom(J_{U}, Id_{V})(A)}
= {A ∈ Hom(U, V) : J_{V} ∘ A = A ∘ J_{U}}

has a canonical CSO, induced by either one of the CSOs, so we may denote the space Hom_{c}(U, V). This is exactly the set of c-linear maps U → V as in Section 5.2. The CSOs on Hom(U, V) induce opposite CSOs on the subspace of a-linear maps,

Hom_{a}(U, V) = {A ∈ Hom(U, V) : Hom(Id_{U}, J_{V})(A) = −Hom(J_{U}, Id_{V})(A)}
= {A ∈ Hom(U, V) : J_{V} ∘ A = −A ∘ J_{U}}.

Example 5.70. Given V = (V, J_{V}), there is a direct sum

End(V) = End_{c}(V) ⊕ End_{a}(V)

as in Example 5.69, where End_{c}(V) admits a canonical CSO. The identity element Id_{V} ∈ End_{c}(V) ⊆ End(V) is c-linear, and so is J_{V}.

Lemma 5.71. For c-linear maps A : U′ → U and B : V → V′, the map

Hom(A, B) : Hom(U, V) → Hom(U′, V′)

respects the direct sums from Example 5.69. The induced map

P′_{c} ∘ Hom(A, B) ∘ Q_{c} : Hom_{c}(U, V) → Hom_{c}(U′, V′) : F ↦ B ∘ F ∘ A,

is c-linear.

Proof. P′_{c} is the projection Hom(U′, V′) ↠ Hom_{c}(U′, V′). Lemma 5.57 applies.

Notation 5.72. The induced map from Lemma 5.71 is denoted

Hom_{c}(A, B) = P′_{c} ∘ Hom(A, B) ∘ Q_{c} : Hom_{c}(U, V) → Hom_{c}(U′, V′).

As in Notation 5.67, this is a restriction of the map Hom(A, B) to subspaces of its domain and target.

Exercise 5.73. For a-linear maps A : U′ → U and B : V → V′, the map

Hom(A, B) : Hom(U, V) → Hom(U′, V′)

respects the direct sums. The induced map

P′_{c} ∘ Hom(A, B) ∘ Q_{c} : Hom_{c}(U, V) → Hom_{c}(U′, V′) : F ↦ B ∘ F ∘ A

is a-linear.

Example 5.74. Given U and V, suppose there are commuting CSOs J_{1}, J_{2} on V. Then the CSOs Hom(Id_{U}, J_{1}) and Hom(Id_{U}, J_{2}) on Hom(U, V) also commute. As in Example 5.51, this gives a direct sum temporarily denoted

(5.8)  Hom(U, V) = (Hom(U, V))_{c} ⊕ (Hom(U, V))_{a}

with projections

\frac{1}{2} · (Id_{Hom(U,V)} ± Hom(Id_{U}, J_{1}) ∘ Hom(Id_{U}, J_{2})).

Hom(U, V) also admits a direct sum Hom(U, V_{c}) ⊕ Hom(U, V_{a}) as in Example 1.78, with projections

(5.9)  Hom(Id_{U}, \frac{1}{2} · (Id_{V} ± J_{1} ∘ J_{2})).

The pairs of projections are identical: this is a special case of Example 1.137. We can identify Hom(U, V_{c}) = (Hom(U, V))_{c}, and also identify Hom(Id_{U}, Q_{c}) with the inclusion of the subspace (Hom(U, V))_{c} in Hom(U, V); similarly, (Hom(U, V))_{a} = Hom(U, V_{a}). More specifically, the set (Hom(U, V))_{c} is defined as {A ∈ Hom(U, V) : J_{1} ∘A = J_{2} ∘A}, while elements of Hom(U, V_{c}) are maps A such that for any u ∈ U, A(u) ∈ V_{c} ⊆ V, meaning J_{1}(A(u)) = J_{2}(A(u)).

Example 5.75. Given U and V, suppose there are commuting CSOs J_{1}, J_{2} on V. Then the CSOs Hom(J_{1}, Id_{U}) and Hom(J_{2}, Id_{U}) on Hom(V, U) also commute. As in Example 5.51, this gives a direct sum Hom(V, U) = (Hom(V, U))_{c} ⊕ (Hom(V, U))_{a} (the same notation as (5.8) but not the same subspaces) with projections

\frac{1}{2} · (Id_{Hom(V,U)} ± Hom(J_{1}, Id_{U}) ∘ Hom(J_{2}, Id_{U})).

Hom(V, U) also admits a direct sum Hom(V_{c}, U) ⊕ Hom(V_{a}, U) as in Example 1.79, with projections Hom(Q_{c}, Id_{U}), Hom(Q_{a}, Id_{U}). Unlike Examples 5.64 and 5.74, these pairs of projections are not obviously identical, and in fact this is a special case of Example 1.138. The set (Hom(V, U))_{c} is defined as {A ∈ Hom(V, U) : A ∘ J_{1} = A ∘ J_{2}}, while elements of Hom(V_{c}, U) are maps A defined only on the subspace of v ∈ V such that J_{1}(v) = J_{2}(v). The two direct sums are different but equivalent, as discussed in Example 1.138. Specifically, if, for i = c, a, P′′_{i}, Q′′_{i} denote the projections and inclusions for the direct sum Hom(V, U) = (Hom(V, U))_{c} ⊕(Hom(V, U))_{a}, then

Q′′_{i} ∘ P′′_{i} = Hom(P_{i}, Id_{U}) ∘ Hom(Q_{i}, Id_{U}) : Hom(V, U) → Hom(V, U),

and as in Lemma 1.95,

P′′_{i} ∘ Hom(P_{i}, Id_{U}) : Hom(V_{i}, U) → (Hom(V, U))_{i}

is invertible with inverse Hom(Q_{i}, Id_{U}) ∘ Q′′_{i}, and for i = c, c-linear.

Lemma 5.76. Given U and V, with commuting CSOs J_{1}, J_{2} ∈ End(V), let W be a space admitting a direct sum W_{1} ⊕ W_{2}. If H : W → Hom(V, U) respects one of the two direct sums from Example 5.75, then H also respects the other direct sum. If, further, H is invertible, then both induced maps W_{1} → (Hom(V, U))_{c} and W_{1} → Hom(V_{c}, U) are also invertible. If the direct sum on W is given by commuting CSOs J_{W}, J′_{W}, and H satisfies both Hom(J_{1}, Id_{U}) ∘ H = H ∘ J_{W} and Hom(J_{2}, Id_{U}) ∘ H = H ∘ J′_{W}, then H respects the direct sums and the induced maps are c-linear.

Proof. The claims follow from Lemmas 1.85, 1.94, and 5.57. If the projection and inclusion operators induced on W = W_{c} ⊕ W_{a} by J_{W}, J′_{W} are P′_{i}, Q′_{i}, then the induced maps Hom(Q_{c}, Id_{U}) ∘ H ∘ Q′_{c} : W_{c} → Hom(V_{c}, U) and P′′_{c} ∘ H ∘ Q′_{c} : W_{c} → (Hom(V, U))_{c} are related by composition with the c-linear invertible map from the above Example:

Hom(Q_{c}, Id_{U}) ∘ H ∘ Q′_{c} = (Hom(Q_{c}, Id_{U}) ∘ Q′′_{c}) ∘ (P′′_{c} ∘ H ∘ Q′_{c}).

Exercise 5.77. The results of the previous Lemma have analogues for a map Hom(V, U) → W.

Example 5.78. For U and commuting CSOs J_{1}, J_{2} on V as in Example 5.74, suppose there are also commuting CSOs J′_{1}, J′_{2} on V′. If B : V → V′ respects the direct sums V_{c}⊕V_{a} → V′_{c}⊕V′_{a} (equivalently, B∘J_{1}∘J_{2} = J′_{1}∘J′_{2}∘B by Lemmas 1.120 and 5.52), then for i = c, a, there are induced maps P′_{i}∘B∘Q_{i} : V_{i} → V′_{i}. By Lemma 1.88, for any map A : W → U, the map Hom(A, B) : Hom(U, V) → Hom(W, V′) respects the direct sums

Hom(U, V_{c}) ⊕ Hom(U, V_{a}) → Hom(W, V′_{c}) ⊕ Hom(W, V′_{a})

from Example 5.74, and for i = c, a, the induced map

Hom(Id_{W} , P′_{i}) ∘ Hom(A, B) ∘ Hom(Id_{U}, Q_{i})

is equal to Hom(A, P′_{i} ∘ B ∘ Q_{i}).

Example 5.79. For U, V, V′, A as in Example 5.78, if B : V → V′ is clinear with respect to both pairs J_{1}, J′_{1} and J_{2}, J′_{2}, then B satisfies the hypothesis from Example 5.78, and by Lemma 5.25, Hom(A, B) is also c-linear with respect to the pairs Hom(Id_{U}, J_{1}), Hom(Id_{W} , J′_{1}) and Hom(Id_{U}, J_{2}), Hom(Id_{W} , J′_{2}). Lemma 5.57 applies, so the induced maps from Example 5.78, P′_{c} ∘ B ∘ Q_{c} : V_{c} → V′_{c}, and also Hom(A, P′_{c} ∘ B ∘ Q_{c}), are both c-linear.

Exercise 5.80. Given U, V, W, with U = (U, J_{U}) and V = (V, J_{V}), the map t_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W), Hom(U, W)) respects the direct sums and the induced map

Hom_{c}(U, V) → Hom_{c}(Hom(V, W), Hom(U, W))

is c-linear.

Hint. Exercise 5.36, Exercise 5.37, and then Lemma 5.57 apply.

Example 5.81. For W = ℝ in the previous Exercise, t_{UV}^{ℝ} = t_{UV} : Hom(U, V) → Hom(V^{∗}, U^{∗}). V^{∗} has a CSO J_{V}^{∗} as in Example 5.5, and similarly J_{U}^{∗} is a CSO for U^{∗}. t_{UV} respects the direct sums

Hom_{c}(U, V) ⊕ Hom_{a}(U, V) → Hom_{c}(V^{∗}, U^{∗}) ⊕ Hom_{a}(V^{∗}, U^{∗})

and the induced map Hom_{c}(U, V) → Hom_{c}(V^{∗}, U^{∗}), A ↦ t_{UV} (A) = Hom(A, Id_{ℝ}) = A^{∗}, is c-linear.

Exercise 5.82. Given U, V, W, with U = (U, J_{U}) and V = (V, J_{V}), the map e_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W) ⊗ U, W) respects the direct sums and the induced map

Hom_{c}(U, V) → Hom(Hom(V, W) ⊗_{c} U, W)

is c-linear.

Hint. Exercise 5.41, Exercise 5.42, and then Lemma 5.76, apply.

Exercise 5.83. Given U, V, W, with U = (U, J_{U}) and W = (W, J_{W}), the map T_{U,V;W} : Hom(U, Hom(V, W)) → Hom(V, Hom(U, W)) respects the direct sums and the induced map

Hom_{c}(U, Hom(V, W)) → Hom(V, Hom_{c}(U, W))

is c-linear and invertible, where the inverse is induced by T_{V,U;W}.

Hint. The direct sum for the domain is as in Example 5.69, and for the target is as in Example 5.74. Exercise 5.45, Exercise 5.46, and then Lemma 5.57 apply.

Exercise 5.84. Given U, V, W, with U = (U, J_{U}) and V = (V, J_{V}), the map T_{U,V;W} : Hom(U, Hom(V, W)) → Hom(V, Hom(U, W)) respects the direct sums and the induced map

Hom_{c}(U, Hom(V, W)) → Hom_{c}(V, Hom(U, W))

is c-linear and invertible, where the inverse is induced by T_{V,U;W}.

Hint. The direct sums are as in Example 5.69. Exercise 5.46 and Lemma 5.57 apply.

Exercise 5.85. Given W and V = (V, J_{V}), the involution T_{V;W} from Notation 4.3 is c-linear with respect to both induced CSOs as follows:

Hom(V, Hom(V, W)) → Hom(V, Hom(V, W))
Hom(V, Hom(V, W)) → Hom(V, Hom(V, W)).

The induced map

Hom_{c}(V, Hom(V, W)) → Hom_{c}(V, Hom(V, W))

is a c-linear involution, and its fixed point subspace has a canonical CSO.

Hint. The c-linearity claims are the U = V special case of Exercise 5.46, and the induced map is the special case from Exercise 5.84. Lemma 5.63 applies to the involution T_{V;W} and the commuting CSOs on Hom(V, Hom(V, W)).

Example 5.86. For W, V = (V, J_{V}), and Hom(V, Hom(V, W)) as in Exercise 5.85, Example 5.61 applies to the commuting involutions T_{V;W} and

−Hom(J_{V} , Id_{Hom(V,W)})∘Hom(Id_{V} , Hom(J_{V} , Id_{W})) = −Hom(J_{V} , Hom(J_{V} , Id_{W})).

Recalling the direct sums from (4.6) and (5.7),

Hom(V, Hom(V, W)) = Sym(V; W) ⊕ Alt(V; W),
Hom(V, Hom(V, W)) = Hom_{c}(V, Hom(V, W)) ⊕ Hom_{a}(V, Hom(V, W)),

the four distinguished subspaces from Example 5.61 can be described in terms of properties of their elements, W -valued bilinear forms h, and are denoted as follows:

Sym_{c}(V, W) = Sym(V; W) ∩ Hom_{c}(V, Hom(V, W))
= {h : (h(v_{1}))(v_{2}) = (h(v_{2}))(v_{1}) = −(h(J_{V} (v_{1})))(J_{V} (v_{2}))},
Sym_{a}(V, W) = Sym(V; W) ∩ Hom_{a}(V, Hom(V, W))
= {h : (h(v_{1}))(v_{2}) = (h(v_{2}))(v_{1}) = (h(J_{V} (v_{1})))(J_{V} (v_{2}))},
Alt_{c}(V, W) = Alt(V; W) ∩ Hom_{c}(V, Hom(V, W))
= {h : (h(v_{1}))(v_{2}) = −(h(v_{2}))(v_{1}) = −(h(J_{V} (v_{1})))(J_{V} (v_{2}))},
Alt_{a}(V, W) = Alt(V; W) ∩ Hom_{a}(V, Hom(V, W))
= {h : (h(v_{1}))(v_{2}) = −(h(v_{2}))(v_{1}) = (h(J_{V} (v_{1})))(J_{V} (v_{2}))}.

The c-linear involution on Hom_{c}(V, Hom(V, W)) induced by T_{V;W} from Exercise 5.85 produces the direct sum

Hom_{c}(V, Hom(V, W)) = Sym_{c}(V, W) ⊕ Alt_{c}(V, W),

and there is a canonical CSO on Sym_{c}(V, W).

Exercise 5.87. Given V with CSOs J_{1}, J_{2}, if J_{1} and J_{2} commute and J_{2} ≠ ±J_{1}, then the span of {Id_{V} , J_{1}, J_{2}, J_{1} ∘ J_{2}} is a 4-dimensional subspace of End(V). This subspace is closed under composition, so it is a subrng of End(V).

Hint. The first claim follows from Exercise 5.20. Such a subspace is a commutative ring, and isomorphic to the associative algebra of \underline{bicomplex numbers} of C. Segre.

5.3.2. Three Commuting Complex Structure Operators.

Example 5.88. Given V and three commuting CSOs J_{1}, J_{2}, J_{3}, consider an ordered triple (i_{1}, i_{2}, i_{3}) which is a permutation (no repeats) of the indices 1, 2,

3. For the first two indices, the two CSOs J_{i_{1}}, J_{i_{2}} produce a direct sum V = V_{c(i_{1}i_{2})} ⊕ V_{a(i_{1}i_{2})} with projection P_{c(i_{1}i_{2})} = \frac{1}{2} · (Id_{V} − J_{i_{1}} ∘ J_{i_{2}}) as in Example 5.51. The ordering of the pair is irrelevant: V_{c(i_{1}i_{2})} = V_{c(i_{2}i_{1})}. The remaining CSO J_{i_{3}} respects this direct sum by Lemma 5.52, and by Lemma 5.28 induces a CSO P_{c(i_{1}i_{2})} ∘ J_{i_{3}} ∘ Q_{c(i_{1}i_{2})} on V_{c(i_{1}i_{2})}, which commutes with the CSO induced by J_{i_{1}} and J_{i_{2}}. So, again as in Example 5.51, there is a direct sum V_{c(i_{1}i_{2})} = (V_{c(i_{1}i_{2})})_{c} ⊕(V_{c(i_{1}i_{2})})_{a} with projection P_{c((i_{1}i_{2})i_{3})} : V_{c(i_{1}i_{2})} ↠ (V_{c(i_{1}i_{2})})_{c}. Simplifying the composition of projections gives the formula

P_{c((i_{1}i_{2})i_{3})} ∘ P_{c(i_{1}i_{2})} = \frac{1}{4} · (Id_{V} − J_{1} ∘ J_{2} − J_{2} ∘ J_{3} − J_{1} ∘ J_{3}).

Considering (V_{c(i_{1}i_{2})})_{c} as a subspace of V, the above formula shows that neither the composite map nor its image depends on the ordering of the three indices, and so (V_{c(i_{1}i_{2})})_{c} is equal to the subspace where all three CSOs coincide and it can be denoted V_{c(123)}. The composite of inclusions Q_{c(i_{1}i_{2})} ∘ Q_{c((i_{1}i_{2})i_{3})} also does not depend on the ordering. The canonical CSO on V_{c(123)} is:

(5.10)  P_{c((i_{1}i_{2})i_{3})} ∘ P_{c(i_{1}i_{2})} ∘ J_{i} ∘ Q_{c(i_{1}i_{2})} ∘ Q_{c((i_{1}i_{2})i_{3})},

for any i = 1, 2, or 3.

Example 5.88 is also a special case of both Theorem 1.124 and Example 5.61: given three commuting CSOs, there are commuting involutions on V,

(5.11)  K_{1} = −J_{2} ∘ J_{3}, K_{2} = −J_{1} ∘ J_{3}, K_{1} ∘ K_{2} = −J_{1} ∘ J_{2}.

The direct sums from Theorem 1.124 produced by these involutions are exactly V = V_{c(i_{1}i_{2})} ⊕V_{a(i_{1}i_{2})}, and each V_{c(i_{1}i_{2})} admits a canonically induced involution and direct sum. The conclusions of Theorem 1.124 are V_{c(12)} ∩ V_{c(23)} ∩ V_{c(13)} = V_{c(123)} and (V_{c(i_{1}i_{2})})_{a} = V_{a(i_{1}i_{3})} ∩ V_{a(i_{2}i_{3})}. From (1.16), each projection P_{c(i_{1}i_{2})i_{3}} is equal to a map induced by P_{c(i_{1}i_{3})} and also to a map induced by P_{c(i_{2}i_{3})}.

Lemma 5.89. For V with three commuting CSOs J_{1}, J_{2}, J_{3}, U = (U, J_{U}), and a map H : V → U, if H is c-linear with respect to (J_{1}, J_{U}), then H ∘Q_{c(12)} : V_{c(12)} → U is c-linear with respect to (P_{c(12)} ∘ J_{1} ∘ Q_{c(12)}, J_{U}) and H ∘ Q_{c(13)} : V_{c(13)} → U is c-linear with respect to (P_{c(13)} ∘ J_{1} ∘ Q_{c(13)}, J_{U}). For any ordering (i_{1}, i_{2}, i_{3}), H ∘ Q_{c(i_{1}i_{2})} ∘ Q_{c((i_{1}i_{2})i_{3})} : V_{c(123)} → U is c-linear.

Proof. The first claim follows from Lemma 5.55, for the induced CSO P_{c(12)} ∘ J_{1} ∘ Q_{c(12)} = P_{c(12)} ∘ J_{2} ∘ Q_{c(12)} on V_{c(12)}. The second claim similarly follows from Lemma 5.55, and there are analogous claims if H is instead assumed to be c-linear with respect to either (J_{2}, J_{U}) or (J_{3}, J_{U}). Lemma 5.55 then applies to H ∘ Q_{c(12)}, and the two CSOs on V_{c(12)} from Example 5.88, P_{c(12)} ∘ J_{1} ∘ Q_{c(12)} and P_{c(12)} ∘ J_{3} ∘ Q_{c(12)}, so (H ∘ Q_{c(12)}) ∘ Q_{c((12)3)} is c-linear with respect to (P_{c((12)3)} ∘ (P_{c(12)} ∘ J_{1} ∘ Q_{c(12)}) ∘ Q_{c((12)3)}, J_{U}). The last claim follows from the composites not depending on the ordering.

Lemma 5.90. Given V with commuting CSOs J_{V}^{1} , J_{V}^{2} , J_{V}^{3} , and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, J_{U}^{3}, if H : U → V satisfies H ∘J_{U}^{1} = J_{V}^{1} ∘H and H ∘J_{U}^{2} = J_{V}^{2} ∘H and H ∘ J_{U}^{3} = J_{V}^{3} ∘ H, then H respects the corresponding direct sums from Example 5.88, and each induced map P_{c(i_{1}i_{2})}^{V} ∘H ∘Q_{c(i_{1}i_{2})}^{U} is c-linear with respect to the CSOs induced by J_{U}^{i_{1}}, J_{V}^{i_{1}} and also c-linear with respect to the CSOs induced by J_{U}^{i_{3}}, J_{V}^{i_{3}}. The induced map

P_{c((i_{1}i_{2})i_{3})}^{V} ∘ P_{c(i_{1}i_{2})}^{V} ∘ H ∘ Q_{c(i_{1}i_{2})}^{U} ∘ Q_{c((i_{1}i_{2})i_{3})}^{U} : U_{c(123)} → V_{c(123)}

is c-linear and does not depend on the ordering (i_{1}, i_{2}, i_{3}). If, also, H is invertible, then the induced maps are invertible.

Proof. Theorem 1.125 applies, with commuting involutions on both V and U as in Example 5.88. In particular, H respects the direct sums

U_{c(i_{1}i_{2})} ⊕ U_{a(i_{1}i_{2})} → V_{c(i_{1}i_{2})} ⊕ V_{a(i_{1}i_{2})}.

The induced map P_{c(i_{1}i_{2})}^{V} ∘ H ∘ Q_{c(i_{1}i_{2})}^{U} respects the direct sum

U_{c((i_{1}i_{2})i_{3})} ⊕ (U_{c(i_{1}i_{2})})_{a} → V_{c((i_{1}i_{2})i_{3})} ⊕ (V_{c(i_{1}i_{2})})_{a},

where U_{c((i_{1}i_{2})i_{3})} = U_{c(123)} and (U_{c(i_{1}i_{2})})_{a} = U_{a(i_{1}i_{3})} ∩ U_{a(i_{2}i_{3})} as in Example 5.88. The first claim of c-linearity follows from Lemma 5.57. The second c-linearity requires checking

P_{c(i_{1}i_{2})}^{V}∘H∘Q_{c(i_{1}i_{2})}^{U}∘P_{c(i_{1}i_{2})}^{U}∘J_{U}^{i_{3}}∘Q_{c(i_{1}i_{2})}^{U} = P_{c(i_{1}i_{2})}^{V}∘J_{V}^{i_{3}}∘Q_{c(i_{1}i_{2})}^{V}∘P_{c(i_{1}i_{2})}^{V}∘H∘Q_{c(i_{1}i_{2})}^{U},

and then the last claims follow from Lemma 5.57 applied again to the commuting CSOs induced on U_{c(i_{1}i_{2})}, V_{c(i_{1}i_{2})}.

Example 5.91. Given U with commuting CSOs J_{1}, J_{2}, J_{3}, suppose H is an involution on U such that H ∘ J_{1} = J_{2} ∘ H and H ∘ J_{3} = J_{3} ∘ H. By Lemma 5.63, H respects the direct sum U_{c(12)} ⊕ U_{a(12)} → U_{c(12)} ⊕ U_{a(12)}, and induces involutions on U_{c(12)} and U_{a(12)}. Lemma 5.90 applies to the triple (J_{1}, J_{2}, J_{3}) on the domain of H and (J_{2}, J_{1}, J_{3}) on the target. So, H respects the direct sums:

(5.12)  U_{c(13)} ⊕ U_{a(13)} → U_{c(23)} ⊕ U_{a(23)}
U_{c(23)} ⊕ U_{a(23)} → U_{c(13)} ⊕ U_{a(13)},

and the induced maps U_{c(13)} → U_{c(23)} → U_{c(13)} are c-linear and mutually inverses. The induced maps U_{a(13)} → U_{a(23)} → U_{a(13)} are also mutually inverses. The subspace U_{c(12)} admits commuting CSOs P_{c(12)} ∘J_{1} ∘Q_{c(12)} = P_{c(12)} ∘J_{2} ∘Q_{c(12)} and P_{c(12)} ∘ J_{3} ∘ Q_{c(12)} as in Example 5.88, producing the direct sum U_{c(123)} ⊕ (U_{c(12)})_{a}. The induced involution P_{c(12)} ∘ H ∘ Q_{c(12)} on U_{c(12)} commutes with both of these CSOs, respects this direct sum, and induces a c-linear involution on U_{c(123)} by Lemma 5.62. The involutions on U from (5.11) satisfy

(5.13)  (−J_{2} ∘ J_{3}) ∘ H = H ∘ (−J_{1} ∘ J_{3}),

so we have a special case of Example 1.126. Adapting the notation from Example 1.126, the involutions H and H ∘ (−J_{1} ∘ J_{2}) produce direct sums U = U_{7} ⊕ U_{8} and U_{9} ⊕ U_{10}. The involution P_{c(12)} ∘ H ∘ Q_{c(12)} on U_{c(12)} commutes with P_{c(12)} ∘ (−J_{1} ∘ J_{3})∘Q_{c(12)} = P_{c(12)}∘(−J_{2}∘J_{3})∘Q_{c(12)}, and their product P_{c(12)}∘(−J_{1}∘J_{3}∘H)∘Q_{c(12)} is an involution with fixed point subspace U_{11}. The following commutative diagram is adapted from Example 1.126.

\xymatrix{ &  & U \ar@{->>}[dll] \ar@{->>}[dl] \ar@{->>}[d] \ar@{->>}[dr] \ar@{->>}[drr] \\ U_{c(23)} \ar@{->>}[dr] & U_{c(13)} \ar@{->>}[d] & U_{c(12)} \ar@{->>}[dl] \ar@{->>}[d] \ar@{->>}[dr] & U_{7} \ar@{->>}[d] & U_{9} \ar@{->>}[dl] \\  & U_{c(123)} \ar@{->>}[dr] & U_{11} \ar@{->>}[d] & U_{c(12)} ∩ U_{7} ∩ U_{9} \ar@{->>}[dl] \\  &  & U_{c(123)} ∩ U_{7}}

The subspace U_{c(123)} ∩ U_{7} has a canonical CSO.

Exercise 5.92. For J_{1}, J_{2}, J_{3}, and H as in Example 5.91, the sixteen operators

{±Id_{U}, ±H, ±J_{1} ∘ J_{2}, ±J_{1} ∘ J_{3}, ±J_{2} ∘ J_{3}, ±H ∘ J_{1} ∘ J_{2}, ±H ∘ J_{1} ∘ J_{3}, ±H ∘ J_{2} ∘ J_{3}}

form a group which is the image of a representation of D_{4} × ℤ_{2}. Unlike the group from Exercise 1.133, there is no pair of anticommuting elements.

Example 5.93. For U = (U, J_{U}), V = (V, J_{V}), and W = (W, J_{W}), the space U ⊗ V ⊗ W admits three commuting CSOs [J_{U} ⊗ Id_{V ⊗W}], …, [Id_{U⊗V} ⊗ J_{W}]. The subspaces (U ⊗_{c} V) ⊗_{c} W and U ⊗_{c} (V ⊗_{c} W) are equal, as a special case of Example 5.88.

Example 5.94. For U = (U, J_{U}) and commuting CSOs J_{1}, J_{2} on V, U ⊗V has three commuting CSOs: [J_{U} ⊗ Id_{V}], [Id_{U} ⊗ J_{1}], [Id_{U} ⊗ J_{2}]. This is another special case of Example 5.88. The projection P_{c(23)} : U ⊗ V ↠ (U ⊗ V)_{c(23)} = U ⊗ V_{c} is as in Example 5.64, so P_{c(23)} = [Id_{U} ⊗ P_{c}], where P_{c} : V ↠ V_{c} is as in Example 5.51. The subspace where all three CSOs agree is (U ⊗ V)_{c(123)} = U ⊗_{c} V_{c}.

Example 5.95. Given U and V, suppose V = (V, J_{V}) and there are commuting CSOs J_{U}, J′_{U} on U. Then J_{1} = Hom(J_{U}, Id_{V}), J_{2} = Hom(J′_{U}, Id_{V}), J_{3} = Hom(Id_{U}, J_{V}) are three commuting CSOs on Hom(U, V), and this is a special case of Example 5.88. There are three direct sums:

Hom(U, V) = (Hom(U, V))_{c(13)} ⊕ (Hom(U, V))_{a(13)},

where (Hom(U, V))_{c(13)} = Hom_{c}((U, J_{U}), V) as in Example 5.69,

Hom(U, V) = (Hom(U, V))_{c(23)} ⊕ (Hom(U, V))_{a(23)},

where (Hom(U, V))_{c(23)} = Hom_{c}((U, J′_{U}), V), and

Hom(U, V) = (Hom(U, V))_{c(12)} ⊕ (Hom(U, V))_{a(12)},

as in Example 5.75. Each (Hom(U, V))_{c(i_{1}i_{2})} admits a direct sum with projection onto

((Hom(U, V))_{c(i_{1}i_{2})})_{c} = (Hom(U, V))_{c(123)} = {A:U →V : A∘J_{U} = A∘J′_{U} = J_{V} ∘A}.

The invertible map

P′′_{c} ∘ Hom(P_{c}, Id_{U}) : Hom(U_{c}, V) → (Hom(U, V))_{c(12)}

from Example 5.75 (where in this case, P′′_{c} = P_{c(12)}) is c-linear with respect to Hom(Id_{U_{c}}, J_{V}) and P_{c(12)} ∘J_{3} ∘Q_{c(12)}, and is also c-linear with respect to Hom(P_{c} ∘ J_{U} ∘ Q_{c}, Id_{V}) and P_{c(12)} ∘ J_{1} ∘ Q_{c(12)}, so by Lemma 5.57, it respects the direct sums and induces an invertible, c-linear map Hom_{c}(U_{c}, V) → (Hom(U, V))_{c(123)}, as indicated in the following diagram.

\xymatrix{ &  & Hom(U, V) \ar[dll]_{Hom(Q_{c},Id_{V} )} \ar@{->>}[dl]^{P_{c(12)}} \ar@{->>}[d]^{P_{c(13)}} \ar@{->>}[dr]^{P_{c(23)}} \\ Hom(U_{c}, V) \ar[urr]_{Hom(P_{c},Id_{V} )} \ar[r] \ar@{->>}[d] & (Hom(U, V))_{c(12)} \ar@{->>}[dr] \ar[l] & (Hom(U, V))_{c(13)} \ar@{->>}[d] & (Hom(U, V))_{c(23)} \ar@{->>}[dl] \\ Hom_{c}(U_{c}, V) \ar[rr] &  & (Hom(U, V))_{c(123)} \ar[ll]}

Lemma 5.96. Given V = (V, J_{V}) and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, let W be a space with three commuting CSOs J_{W}^{1} , J_{W}^{2} , J_{W}^{3} . If H : W → Hom(U, V) satisfies Hom(J_{U}^{1}, Id_{V}) ∘ H = H ∘ J_{W}^{1} and Hom(J_{U}^{2}, Id_{V}) ∘ H = H ∘ J_{W}^{2} and Hom(Id_{U}, J_{V}) ∘ H = H ∘ J_{W}^{3} , then H respects the corresponding direct sums from Example 5.95 and the induced maps are c-linear. If also H is invertible, then the induced maps are invertible.

Proof. That H respects the direct sums produced by the three corresponding pairs of CSOs, and that the induced maps

P_{c(i_{1}i_{2})} ∘ H ∘ Q′_{c(i_{1}i_{2})} : W_{c(i_{1}i_{2})} → (Hom(U, V))_{c(i_{1}i_{2})}

(for example, the arrow labeled a_{3} in the diagram below) are c-linear, follow from Lemma 5.90, which also showed the induced map ã_{3} : W_{c(123)} → (Hom(U, V))_{c(123)} is c-linear, and invertible if H is. In the diagram, a_{1} and ã_{1} are the canonical invertible maps which appeared as horizontal arrows in the diagram from Example 5.95; the adjacent projection arrows are also copied from that diagram. Lemma 5.76 showed that H also respects the direct sum W_{c(12)} ⊕ W_{a(12)} → Hom(U_{c}, V) ⊕ Hom(U_{a}, V) and that the induced map Hom(Q_{c}, Id_{V})∘H ∘Q′_{c(12)}, denoted a_{2} in the diagram below, is c-linear with respect to the CSO induced by J_{W}^{1} and the CSO induced by Hom(J_{U}^{1}, Id_{V}). In fact, a_{2} is also c-linear with respect to the other pair of CSOs, induced by J_{W}^{3} and Hom(Id_{U}, J_{V}). By Lemma 5.57, a_{2} respects the direct sums produced by the commuting CSOs and induces a c-linear map ã_{2} : W_{c(123)} → Hom_{c}(U_{c}, V); it satisfies the identity ã_{2} = ã_{1} ∘ ã_{3}.

[figure: Hom(U, V); P_{c(12)}; Hom(Q_{c},Id_{V} ); W; P′_{c(12)}; H; W_{c(12)}; a_{3}; a_{2}; P′_{c((12)3)}; (Hom(U, V))_{c(12)}; a_{1}; P_{c((12)3)}; Hom(U_{c}, V); W_{c(123)}; ã_{3}; ã_{2}; (Hom(U, V))_{c(123)}; ã_{1}; Hom_{c}(U_{c}, V)]

Example 5.97. Given U and V, suppose there are three commuting CSOs J_{U}, J′_{U}, J′′_{U} on U. Then J_{1} = Hom(J_{U}, Id_{V}), J_{2} = Hom(J′_{U}, Id_{V}), J_{3} = Hom(J′′_{U}, Id_{V}) are three commuting CSOs on Hom(U, V), and this is a special case of Example 5.88. There are three direct sums: Hom(U, V) = (Hom(U, V))_{c(i_{1}i_{2})} ⊕(Hom(U, V))_{a(i_{2}i_{2})}, each of which is equivalent to a direct sum Hom(U_{c(i_{1}i_{2})}, V)⊕Hom(U_{a(i_{1}i_{2})}, V) as in Example 5.75, with projection Hom(Q_{c(i_{2},i_{2})}, Id_{V}) : Hom(U, V) → Hom(U_{c(i_{1}i_{2})}, V) and inclusion Hom(P_{c(i_{2},i_{2})}, Id_{V}). Each (Hom(U, V))_{c(i_{1}i_{2})} admits a direct sum with projection P′_{c((i_{1}i_{2})i_{3})} onto

((Hom(U, V))_{c(i_{1}i_{2})})_{c} = (Hom(U, V))_{c(123)} = {A: U → V : A∘J_{U} = A∘J′_{U} = A∘J′′_{U}}.

Each subspace Hom(U_{c(i_{2}i_{2})}, V) has two commuting CSOs, and Example 5.75 applies again; there are equivalent direct sums:

Hom(U_{c(i_{1}i_{2})}, V) = (Hom(U_{c(i_{1}i_{2})}, V))_{c} ⊕ (Hom(U_{c(i_{1}i_{2})}, V))_{a},
Hom(U_{c(i_{1}i_{2})}, V) = Hom(U_{c(123)}, V) ⊕ Hom((U_{c(i_{1}i_{2})})_{a}, V).

The following diagram shows the (i_{1}i_{2}) = (12) case, the other two cases being similar.

\xymatrix{ &  & Hom(U, V) \ar[dl]_{Hom(Q_{c(12)},Id_{V} )} \ar@{->>}[d]^{P′_{c(12)}} \\  & Hom(U_{c(12)}, V) \ar@{-->}[ur] \ar[r]^{a_{1}} \ar@{->>}[d]^{P′′_{c}} \ar[dl]_{Hom(Q_{c((12)3)},Id_{V} )} & (Hom(U, V))_{c(12)} \ar@{->>}[d]^{P′_{c((12)3)}} \\ Hom(U_{c(123)}, V) \ar[r]^{a_{2}} \ar[ur] & (Hom(U_{c(12)}, V))_{c} \ar[r]^{a_{3}} & (Hom(U, V))_{c(123)}}

The horizontal arrows are

a_{1} = P′_{c(12)} ∘ Hom(P_{c(12)}, Id_{V})
a_{2} = P′′_{c} ∘ Hom(P_{c((12)3)}, Id_{V})
a_{3} = P′_{c((12)3)} ∘ a_{1} ∘ Q′′_{c}.

Both a_{1} and a_{2} are c-linear and invertible, canonically induced from the equivalent direct sums as in Example 5.75. The a_{1} map is also c-linear with respect to the CSOs induced by J′′_{U}, so the induced map a_{3} is c-linear and invertible by Lemma 5.57. The c-linear invertible composite

a_{3} ∘ a_{2} = P′_{c((12)3)} ∘ P′_{c(12)} ∘ Hom(P_{c((12)3)} ∘ P_{c(12)}, Id_{V})

is canonical, not depending on the choice of (i_{1}i_{2}), as in Example 5.88.

Example 5.98. Given U and V, suppose U = (U, J_{U}) and there are commuting CSOs J_{V}, J′_{V} on V. Then J_{1} = Hom(J_{U}, Id_{V}), J_{2} = Hom(Id_{U}, J_{V}), J_{3} = Hom(Id_{U}, J′_{V}) are three commuting CSOs on Hom(U, V), and this is a special case of Example 5.88. There are three direct sums:

Hom(U, V) = (Hom(U, V))_{c(12)} ⊕ (Hom(U, V))_{a(12)},

where (Hom(U, V))_{c(12)} = Hom_{c}(U, (V, J_{V})) as in Example 5.69,

Hom(U, V) = (Hom(U, V))_{c(13)} ⊕ (Hom(U, V))_{a(13)},

where (Hom(U, V))_{c(13)} = Hom_{c}(U, (V, J′_{V})), and

Hom(U, V) = (Hom(U, V))_{c(23)} ⊕ (Hom(U, V))_{a(23)},

where (Hom(U, V))_{c(23)} = Hom(U, V_{c}) as in Example 5.74. Each (Hom(U, V))_{c(i_{1}i_{2})} admits a direct sum with projection onto

((Hom(U, V))_{c(i_{1}i_{2})})_{c} = (Hom(U, V))_{c(123)} = Hom_{c}(U, V_{c}),

as follows:

There are two ways to construct the projection

P^{1} : (Hom(U, V))_{c(23)} = Hom(U, V_{c}) ↠ (Hom(U, V))_{c(123)} = Hom_{c}(U, V_{c}),

which will turn out to give the same map. Denote the projection P′_{c} : V ↠ V_{c} as in Example 5.51 with corresponding inclusion Q′_{c}; then P_{c(23)} = Hom(Id_{U}, P′_{c}) : Hom(U, V) ↠ Hom(U, V_{c}) is the projection from (5.9) in Example 5.74, with corresponding inclusion Q_{c(23)} = Hom(Id_{U}, Q′_{c}).

The first construction of P^{1} is to consider Hom(U, V_{c}) as a space with commuting CSOs Hom(J_{U}, Id_{V_{c}}), Hom(Id_{U}, J_{V_{c}}) and directly apply Example 5.69 to get a projection

P^{1} = \frac{1}{2} · (Id_{Hom(U,V_{c})} − Hom(J_{U}, Id_{V_{c}}) ∘ Hom(Id_{U}, J_{V_{c}}))
(5.14)  = \frac{1}{2} · (Id_{Hom(U,V_{c})} − Hom(J_{U}, P′_{c} ∘ J_{V} ∘ Q′_{c})).

Second, consider the subspace (Hom(U, V))_{c(23)}, with two induced CSOs that commute, as in Example 5.88:

(P_{c(23)} ∘ Hom(J_{U}, Id_{V}) ∘ Q_{c(23)}), (P_{c(23)} ∘ Hom(Id_{U}, J_{V}) ∘ Q_{c(23)}).

Then, using P_{c(23)} = Hom(Id_{U}, P′_{c}) and Q_{c(23)} = Hom(Id_{U}, Q′_{c}), the projection P_{c((23)1)} defined by these two CSOs as in Example 5.88 is the same as (5.14).

The projection P_{c((12)3)} : (Hom(U, V))_{c(12)} ↠ (Hom(U, V))_{c(123)} can also be defined by two methods with the same result (and similarly for P_{c((13)2)}). The commuting induced CSOs:

(5.15) (P_{c(12)} ∘ Hom(J_{U}, Id_{V}) ∘ Q_{c(12)}), (P_{c(12)} ∘ Hom(Id_{U}, J′_{V}) ∘ Q_{c(12)})

define P_{c((12)3)} as in Example 5.88. The other way to define the projection is to consider the map Hom(Id_{U}, P′_{c}) : Hom(U, V) ↠ Hom(U, V_{c}), which is c-linear in two different ways: with respect to the pair Hom(J_{U}, Id_{V}), Hom(J_{U}, Id_{V_{c}}) and also the pair Hom(Id_{U}, J_{V}), Hom(Id_{U}, J_{V_{c}}), as in Lemma 5.71. The induced map P^{2} = P^{1} ∘ Hom(Id_{U}, P′_{c}) ∘ Q_{c(12)} : (Hom(U, V))_{c(12)} → Hom_{c}(U, V_{c}) is c-linear; it can be denoted Hom_{c}(Id_{U}, P′_{c}) as in Notation 5.72. By the equality of the composite projections from Example 5.88,

(5.16)  P^{2} = P^{1} ∘ Hom(Id_{U}, P′_{c}) ∘ Q_{c(12)}
(5.17)  = P_{c((23)1)} ∘ P_{c(23)} ∘ Q_{c(12)}
= P_{c((12)3)} ∘ P_{c(12)} ∘ Q_{c(12)} = P_{c((12)3)}.

The expression (5.17) is an example of the construction (1.16) from Theorem 1.124. Similarly, since the composite inclusions are equal:

(5.18)  Q_{c(12)} ∘ Q_{c((12)3)} = Hom(Id_{U}, Q′_{c}) ∘ Q_{c((23)1)},

the inclusion Q_{c((12)3)} is equal the induced map:

Q_{c((12)3)} = P_{c(12)} ∘ Hom(Id_{U}, Q′_{c}) ∘ Q_{c((23)1)} = Hom_{c}(Id_{U}, Q′_{c}).

Exercise 5.99. Given V = (V, J_{V}) and W = (W, J_{W}), Hom(Hom(V, W), W) admits three commuting CSOs,

J_{1} = Hom(Hom(J_{V} , Id_{W}), Id_{W}),
J_{2} = Hom(Hom(Id_{V} , J_{W}), Id_{W}),
J_{3} = Hom(Id_{Hom(V,W)}, J_{W}).

As in Example 5.95, there are three direct sums; (Hom(Hom(V, W), W))_{c(23)} was considered in Exercise 5.40. The composite

P_{c((23)1)} ∘ P_{c(23)} ∘ d_{V W} : V → (Hom(Hom(V, W), W))_{c(123)}

is c-linear with respect to the canonical CSO. Let Q_{c} : Hom_{c}(V, W) ↪ Hom(V, W) denote the inclusion from Example 5.69. Then the image of

Hom(Q_{c}, Id_{W}) ∘ d_{V W} : V → Hom(Hom_{c}(V, W), W)

is contained in Hom_{c}(Hom_{c}(V, W), W), i.e., for any v ∈ V,

d_{V W} (v) ∘ Q_{c} : Hom_{c}(V, W) → W : H ↦ (Q_{c}(H))(v) = H(v)

is a c-linear map. From the commutativity of the diagram from Example 5.95, considering Hom(Q_{c}, Id_{W}) ∘ d_{V W} as a map V → Hom_{c}(Hom_{c}(V, W), W), it is identical to the composite of the above map P_{c((23)1)}∘P_{c(23)}∘d_{V W} with the canonical map (Hom(Hom(V, W), W))_{c(123)} → Hom_{c}(Hom_{c}(V, W), W), so it is c-linear.

Recall from Definition 1.39 and Notation 1.40 the canonical map

n : V ⊗ Hom(U, W) → Hom(U, V ⊗ W) : (n′(v ⊗ E)) : u ↦ v ⊗ (E(u)).

Theorem 5.100. If U = (U, J_{U}) and V = (V, J_{V}) and W = (W, J_{W}), then n : V ⊗ Hom(U, W) → Hom(U, V ⊗ W) is c-linear with respect to corresponding pairs of the three commuting CSOs induced on each space, so it respects the direct sums and induces c-linear maps

n^{1} : V ⊗ Hom_{c}(U, W) → Hom_{c}(U, V ⊗ W)
n^{2} : V ⊗_{c} Hom(U, W) → Hom_{c}(U, V ⊗ W)
n^{3} : V ⊗_{c} Hom(U, W) → Hom(U, V ⊗_{c} W)
n : V ⊗_{c} Hom_{c}(U, W) → Hom_{c}(U, V ⊗_{c} W),

which are invertible if n is.

Proof. The c-linearity claims for n follow from Lemma 1.41 (adjusted for variations in ordering; they are also straightforward to check directly), and then Lemma 5.90 applies. The direct sums for the domain V ⊗ Hom(U, W) are as in Example 5.94. The projection onto the subspace V ⊗ Hom_{c}(U, W) is equal to [Id_{V} ⊗ P_{H}] as in Example 5.64, where P_{H} is the projection Hom(U, W) ↠ Hom_{c}(U, W). The direct sums for the target space Hom(U, V ⊗ W) are as in Example 5.98. The following diagram shows some of the canonical projections, including P^{2} as in (5.16).

\xymatrix{V ⊗ Hom(U, W) \ar[r]^{n} \ar@{->>}[d]_{[Id_{V} ⊗P_{H}]} & Hom(U, V ⊗ W) \ar@{->>}[d] \\ V ⊗ Hom_{c}(U, W) \ar@{->>}[d] \ar[r]^{n^{1}} & Hom_{c}(U, V ⊗ W) \ar@{->>}[d]^{P^{2}=Hom_{c}(Id_{U} ,P′_{c})} \\ V ⊗_{c} Hom_{c}(U, W) \ar[r]^{n} & Hom_{c}(U, V ⊗_{c} W)}

The invertibility also follows from Lemma 5.90; in particular, if U or V is finitedimensional then these maps are invertible.

Example 5.101. Given U = (U, J_{U}), V = (V, J_{V}), W = (W, J_{W}), the canonical invertible map

q : Hom(V, Hom(U, W)) → Hom(V ⊗ U, W)

from Definition 1.44 is c-linear with respect to the three corresponding pairs of induced CSOs, by Lemma 1.47. Lemma 5.96 applies, so that the following induced maps are c-linear and invertible:

Hom_{c}(V, Hom(U, W)) → Hom(V ⊗_{c} U, W)
Hom(V, Hom_{c}(U, W)) → Hom_{c}(V ⊗ U, W)
Hom_{c}(V, Hom(U, W)) → Hom_{c}(V ⊗ U, W)
Hom_{c}(V, Hom_{c}(U, W)) → Hom_{c}(V ⊗_{c} U, W).

The direct sums for the domain Hom(V, Hom(U, W)) are as in Example 5.98. The direct sums for the target space Hom(V ⊗ U, W) are as in Example 5.95.

Example 5.102. Given V = (V, J_{V}) and W = (W, J_{W}), Hom(V, Hom(V, W)) admits three commuting CSOs, and the involution T_{V;W} is c-linear with respect to three corresponding pairs of commuting CSOs as in Exercise 5.45 and Exercise 5.46. Hom(V, Hom(V, W)) also admits the three involutions from (5.11), and the involution T_{V;W} satisfies

(−Hom(Id_{V} , Hom(J_{V} , J_{W}))) ∘ T_{V;W} = T_{V;W} ∘ (−Hom(J_{V} , Hom(Id_{V} , J_{W}))),

as in (5.13), so this is a special case of Example 5.91. As in (5.12), T_{V;W} induces c-linear invertible maps:

Hom_{c}(V, Hom(V, W)) → Hom(V, Hom_{c}(V, W))
Hom(V, Hom_{c}(V, W)) → Hom_{c}(V, Hom(V, W)).

The subspace Hom_{c}(V, Hom(V, W)) admits commuting CSOs as in (5.15), and projects onto the subspace Hom_{c}(V, Hom_{c}(V, W)) as in Example 5.98. The involution on Hom_{c}(V, Hom(V, W)) induced by T_{V;W} from Exercise 5.85 induces a c-linear involution on Hom_{c}(V, Hom_{c}(V, W)), producing a direct sum denoted Sym_{c}(V; W) ⊕ Alt_{c}(V; W). Combining the notation from Example 1.126, Example 5.86, and Example 5.91, the following commutative diagram shows some of the projections from Example 5.91.

\xymatrix{ & Hom(V, Hom(V, W)) \ar@{->>}[dl] \ar@{->>}[d] \ar@{->>}[dr] \\ Hom_{c}(V, Hom(V, W)) \ar@{->>}[d] & Hom_{c}(V, Hom(V, W)) \ar@{->>}[dl]^{P^{2}} \ar@{->>}[d] \ar@{->>}[dr] & Sym(V; W) \ar@{->>}[d] \\ Hom_{c}(V, Hom_{c}(V, W)) \ar@{->>}[dr] & U_{11} \ar@{->>}[d] & Sym_{c}(V; W) \ar@{->>}[dl] \\  & Sym_{c}(V; W)}

The projection P^{2} = P_{c((12)3)} labeled in the diagram, and the vertical arrow P_{c((13)2)} on the left, are as in (5.16) from Example 5.98. The subspace U_{11} is this fixed point subspace:

{h : (h(v_{1}))(v_{2}) = −(h(J_{V} (v_{1})))(J_{V} (v_{2})) = −J_{W} ((h(J_{V} (v_{2})))(v_{1}))}.

The conclusions from Example 5.91 are that

Sym_{c}(V; W) = Hom_{c}(V, Hom_{c}(V, W)) ∩ Sym(V; W),

and that Sym_{c}(V; W) has a canonical CSO.

Exercise 5.103. Given U = (U, J_{U}) and W = (W, J_{W}), Hom(Hom(V, W) ⊗ U, W) admits three commuting CSOs, as in Example 5.95, so there are three direct sums; (Hom(Hom(V, W) ⊗ U, W))_{c(23)} was considered in Exercise 5.43. The composite

P_{c((23)1)} ∘ P_{c(23)} ∘ e_{UV}^{W} : Hom(U, V) → (Hom(Hom(V, W) ⊗ U, W))_{c(123)}

is c-linear with respect to the canonical CSO. Let

Q_{c} : Hom(V, W) ⊗_{c} U ↪ Hom(V, W) ⊗ U

denote the inclusion from Example 5.65. Then the image of

Hom(Q_{c}, Id_{W}) ∘ e_{UV}^{W} : Hom(U, V) → Hom(Hom(V, W) ⊗_{c} U, W)

is contained in Hom_{c}(Hom(V, W) ⊗_{c} U, W), i.e., for any A ∈ Hom(U, V),

e_{UV}^{W} (A) ∘ Q_{c} : Hom(V, W) ⊗_{c} U → W : B ⊗ u ↦ (e_{UV}^{W} (A))(Q_{c}(B ⊗ u)) = B(A(u))

is a c-linear map. From the commutativity of the diagram from Example 5.95, considering Hom(Q_{c}, Id_{W}) ∘ e_{UV}^{W} as a map Hom(U, V) → Hom_{c}(Hom(V, W) ⊗_{c} U, W), it is identical to the composite of the above map P_{c((23)1)} ∘P_{c(23)} ∘e_{UV}^{W} with the canonical map (Hom(Hom(V, W)⊗U, W))_{c(123)} → Hom_{c}(Hom(V, W)⊗_{c}U, W), so it is c-linear.

Exercise 5.104. Given V = (V, J_{V}) and W = (W, J_{W}), Hom(Hom(V, W) ⊗ U, W) admits three commuting CSOs, as in Example 5.95, so there are three direct sums; (Hom(Hom(V, W) ⊗ U, W))_{c(23)} was considered in Exercises 5.43, 5.103. The composite

P_{c((23)1)} ∘ P_{c(23)} ∘ e_{UV}^{W} : Hom(U, V) → (Hom(Hom(V, W) ⊗ U, W))_{c(123)}

is c-linear with respect to the canonical CSO. Let Q_{c} : Hom_{c}(V, W) ⊗ U → Hom(V, W) ⊗ U denote the inclusion from Examples 5.64 and 5.69. Then the image of

Hom(Q_{c}, Id_{W}) ∘ e_{UV}^{W} : Hom(U, V) → Hom(Hom_{c}(V, W) ⊗ U, W)

is contained in Hom_{c}(Hom_{c}(V, W) ⊗ U, W), i.e., for any A ∈ Hom(U, V),

e_{UV}^{W} (A) ∘ Q_{c} : Hom_{c}(V, W) ⊗ U → W : B ⊗ u ↦ (e_{UV}^{W} (A))(Q_{c}(B ⊗ u)) = B(A(u))

is a c-linear map. From the commutativity of the diagram from Example 5.95, considering Hom(Q_{c}, Id_{W}) ∘ e_{UV}^{W} as a map Hom(U, V) → Hom_{c}(Hom_{c}(V, W) ⊗ U, W), it is identical to the composite of the above map P_{c((23)1)} ∘ P_{c(23)} ∘ e_{UV}^{W} with the canonical map (Hom(Hom(V, W)⊗U, W))_{c(123)} → Hom_{c}(Hom_{c}(V, W)⊗U, W), so it is c-linear.

Exercise 5.105. Given U, V, W, with U = (U, J_{U}), V = (V, J_{V}), and W = (W, J_{W}), we consider the three commuting CSOs on Hom(V, W) ⊗ U, so Example 5.97 applies to Hom(Hom(V, W) ⊗ U, W). The lower square in the following commutative diagram is a specific case of a square from the diagram in Example 5.97, with the same labeling of arrows, and (12) referring to the CSOs induced by J_{U} and J_{V}. The e_{1}, e_{2} arrows are the maps induced by e_{UV}^{W} on the two equivalent direct sums from Exercise 5.82.

\xymatrix{ & Hom_{c}(U, V) \ar[dl]_{e_{1}} \ar[d]^{e_{2}} \\ Hom(Hom(V, W) ⊗_{c} U, W) \ar[d]_{Hom(Q_{c((12)3)},Id_{W})} \ar[r]^{a_{1}} & (Hom(Hom(V, W) ⊗ U, W))_{c(12)} \ar@{->>}[d]^{P′_{c((12)3)}} \\ Hom(Hom_{c}(V, W) ⊗_{c} U, W) \ar[r]^{a_{3}∘a_{2}} & (Hom(Hom(V, W) ⊗ U, W))_{c(123)}}

The composite P′_{c((12)3)} ∘ e_{2} is c-linear, so the composite Hom(Q_{c((12)3)}, Id_{W}) ∘ e_{1} is also c-linear. As in Exercises 5.43, 5.103, 5.104, the image of Hom(Q_{c((12)3)}, Id_{W}) ∘ e_{1} is contained in Hom_{c}(Hom_{c}(V, W) ⊗ U, W), i.e., if Q′_{c} denotes the inclusion of Hom_{c}(U, V) ↪ Hom(U, V) and Q_{c(12)} ∘ Q_{c((12)3)} is the composite inclusion of Hom_{c}(V, W) ⊗_{c} U ↪ Hom(V, W) ⊗ U, then for any A ∈ Hom_{c}(U, V), the map

(5.19)  e_{UV}^{W} (Q′_{c}(A)) ∘ Q_{c(12)} ∘ Q_{c((12)3)} : Hom_{c}(V, W) ⊗_{c} U → W

is c-linear.

Example 5.106. For V = (V, J_{V}) and W = (W, J_{W}), the space Hom(V, W)⊗V admits three commuting CSOs as in Example 5.94. The composite of the inclusions, Q_{c(12)} ∘ Q_{c((12)3)},

\xymatrix{Hom_{c}(V, W) ⊗_{c} V \ar@{^{(}->}[r]_{Q_{c((12)3)}} & Hom_{c}(V, W) ⊗ V \ar@{^{(}->}[r]_{Q_{c(12)}} & Hom(V, W) ⊗ V}

does not depend on the ordering of the indices, but in the case of the above diagram, Q_{c(12)} = [Q_{c} ⊗ Id_{V}] for Q_{c} : Hom_{c}(V, W) ↪ Hom(V, W), as in Example 5.64.

Define a c-linear evaluation map

(5.20)  Ev_{VW}^{c} = Ev_{V W} ∘ Q_{c(12)} ∘ Q_{c((12)3)} : Hom_{c}(V, W) ⊗_{c} V → W.

This is the restriction of the canonical evaluation from Definition 2.70; it is c-linear by Exercise 5.44 and Lemma 5.89. Considering the formula Ev_{V W} = e_{V V}^{W} (Id_{V}) from Equation (2.12), this construction is also a special case of (5.19) from Exercise 5.105. The domain Hom_{c}(V, W) ⊗_{c} V is spanned by elements of the form P_{c((12)3)}(A ⊗ v) for c-linear maps A : V → W, on which Ev_{VW}^{c} acts as follows:

Ev_{VW}^{c}(P_{c((12)3)}(A ⊗ v))
= (Ev_{V W} ∘ Q_{c(12)} ∘ Q_{c((12)3)} ∘ P_{c((12)3)})(A ⊗ v)
= (Ev_{V W} ∘ [Q_{c} ⊗ Id_{V}] ∘ \frac{1}{2}(Id_{Hom_{c}(V,W)⊗V} − [J_{Hom_{c}(V,W)} ⊗ J_{V}]))(A ⊗ v)
= Ev_{V W} (\frac{1}{2}((Q_{c}(A)) ⊗ v − (Q_{c}(A ∘ J_{V})) ⊗ (J_{V} (v))))
= \frac{1}{2}(Q_{c}(A))(v) − \frac{1}{2}(Q_{c}(A ∘ J_{V}))(J_{V} (v))
(5.21)= A(v) = Ev_{V W} (A ⊗ v),

where in line (5.21), we just forget that A = Q_{c}(A) and A ∘ J_{V} = Q_{c}(A ∘ J_{V}) are c-linear.

Lemma 5.107. For any U, V, W, and c-linear map B : U → W, the following diagram is commutative.

\xymatrix{U \ar[r]^{B} & W \\ Hom_{c}(V, U) ⊗_{c} V \ar[u]^{Ev_{VU}^{c}} \ar[r]_{[Hom_{c}(Id_{V} ,B)⊗_{c}Id_{V} ]} & Hom_{c}(V, W) ⊗_{c} V \ar[u]_{Ev_{VW}^{c}}}

Proof. This is a c-linear version of Lemma 2.72 (the case G = Id_{V}), which states the commutativity of the upper block of this diagram (for any B, not necessarily c-linear).

\xymatrix{U \ar[r]^{B} & W \\ Hom(V, U) ⊗ V \ar[u]^{Ev_{V U}} \ar[r]_{[Hom(Id_{V} ,B)⊗Id_{V} ]} & Hom(V, W) ⊗ V \ar[u]_{Ev_{V W}} \\ Hom_{c}(V, U) ⊗ V \ar@{^{(}->}[u]^{[Q′_{c}⊗Id_{V} ]} \ar[r]_{[Hom_{c}(Id_{V} ,B)⊗Id_{V} ]} & Hom_{c}(V, W) ⊗ V \ar@{^{(}->}[u]_{[Q_{c}⊗Id_{V} ]} \\ Hom_{c}(V, U) ⊗_{c} V \ar@{^{(}->}[u]^{Q′_{c((12)3)}} \ar[r]_{[Hom_{c}(Id_{V} ,B)⊗_{c}Id_{V} ]} & Hom_{c}(V, W) ⊗_{c} V \ar@{^{(}->}[u]_{Q_{c((12)3)}}}

The middle block is commutative by definition of Hom_{c}(Id_{V} , B) for the c-linear maps Id_{V} and B as in Notation 5.72, together with Lemma 1.36. The lower block is commutative by the construction from Notation 5.67. The upward composites in the left and right columns are Ev_{VU}^{c} and Ev_{VW}^{c} as in Example 5.106.

5.3.3. Four Commuting Complex Structure Operators.

Example 5.108. Given V and four commuting CSOs J_{1}, J_{2}, J_{3}, J_{4}, the construction of Example 5.88 shows that for any ordered pair (i_{1}, i_{2}) selected without repeats from the indices 1, 2, 3, 4, there is a direct sum with projection P_{c(i_{1}i_{2})} : V ↠ V_{c(i_{1}i_{2})}, and for a third distinct index, another direct sum with projection P_{c((i_{1}i_{2})i_{3})} : V_{c(i_{1}i_{2})} ↠ V_{c(i_{1}i_{2}i_{3})}. Repeating the process for the remaining, fourth index, the fourth commuting CSO produces a direct sum V_{c(i_{1}i_{2}i_{3})} = (V_{c(i_{1}i_{2}i_{3})})_{c} ⊕ (V_{c(i_{1}i_{2}i_{3})})_{a}, with projection P_{c((i_{1}i_{2}i_{3})i_{4})} : V_{c(i_{1}i_{2}i_{3})} ↠ (V_{c(i_{1}i_{2}i_{3})})_{c}. As in Example 5.88, the composite P_{c((i_{1}i_{2}i_{3})i_{4})} ∘ P_{c((i_{1}i_{2})i_{3})} ∘ P_{c(i_{1}i_{2})} equals

\frac{1}{8} · (Id_{V} − J_{1} ∘ J_{2} − J_{2} ∘ J_{3} − J_{1} ∘ J_{3} − J_{1} ∘ J_{4} − J_{2} ∘ J_{4} − J_{3} ∘ J_{4} + J_{1} ∘ J_{2} ∘ J_{3} ∘ J_{4}),

which shows the image of the last projection does not depend on the ordering of the four indices, so the subspace where all four CSOs coincide can be denoted V_{c(1234)}.

Example 5.109. Given V and four commuting CSOs J_{1}, J_{2}, J_{3}, J_{4}, the space V_{c(i_{1}i_{2})} from Example 5.108 admits three commuting CSOs: P_{c(i_{1}i_{2})} ∘ J_{i_{1}} ∘ Q_{c(i_{1}i_{2})}, P_{c(i_{1}i_{2})} ∘ J_{i_{3}} ∘ Q_{c(i_{1}i_{2})}, P_{c(i_{1}i_{2})} ∘ J_{i_{4}} ∘ Q_{c(i_{1}i_{2})}. Example 5.108 considered pairing the first one with one of the other two to get two direct sums, but as in Example 5.88, there are three possible direct sums on V_{c(i_{1}i_{2})}, the third coming from P_{c(i_{1}i_{2})} ∘J_{i_{3}} ∘ Q_{c(i_{1}i_{2})}, P_{c(i_{1}i_{2})} ∘ J_{i_{4}} ∘ Q_{c(i_{1}i_{2})} to get a subspace denoted V_{c(i_{1}i_{2})(i_{3}i_{4})}, where J_{i_{1}} = J_{i_{2}} and J_{i_{3}} = J_{i_{4}}, equal to the subspace V_{c(i_{3}i_{4})(i_{1}i_{2})}. The composite projection V ↠ V_{c(i_{1}i_{2})(i_{3}i_{4})} is given by the formula

\frac{1}{4} · (Id_{V} − J_{1} ∘ J_{2} − J_{3} ∘ J_{4} + J_{1} ∘ J_{2} ∘ J_{3} ∘ J_{4}).

The subspace V_{c(i_{1}i_{2})(i_{3}i_{4})} admits two commuting CSOs, the one induced by J_{i_{1}} and J_{i_{2}}, and the other by J_{i_{3}} and J_{i_{4}}, so there is a direct sum, and a projection onto the subspace V_{c(1234)} from Example 5.108.

Theorem 5.110. Given V and four commuting CSOs J_{1}, J_{2}, J_{3}, J_{4}, the following diagram is commutative, where the arrows are all the projections from direct sums produced by commuting CSOs described in Examples 5.108 and 5.109.

\xymatrix{ &  &  & V \ar@{->>}[dl] \ar@{->>}[ddll] \ar@{->>}[dddlll] \ar@{->>}[dr] \ar@{->>}[ddrr] \ar@{->>}[dddrrr] \\  &  & V_{c(14)} \ar@{->>}[ddddd] \ar@{->>}[dddddr] \ar@{->>}[dddddrr] &  & V_{c(23)} \ar@{->>}[dddllll] \ar@{->>}[dddddl] \ar@{->>}[dddrr] \\  & V_{c(13)} \ar@{->>}[ddl] \ar@{->>}[ddddrrr] \ar@{->>}[dddrrrr] &  &  &  & V_{c(24)} \ar@{->>}[ddddlll] \ar@{->>}[ddd] \ar@{->>}[ddr] \\ V_{c(12)} \ar@{->>}[d] \ar@{->>}[ddr] \ar@{->>}[dddrr] &  &  &  &  &  & V_{c(34)} \ar@{->>}[ddlllll] \ar@{->>}[dddll] \ar@{->>}[d] \\ V_{c(123)} \ar@{->>}[dddrrr] &  &  &  &  &  & V_{c(234)} \ar@{->>}[dddlll] \\  & V_{c(12)(34)} \ar@{->>}[ddrr] &  &  &  & V_{c(13)(24)} \ar@{->>}[ddll] \\  &  & V_{c(124)} \ar@{->>}[dr] & V_{c(14)(23)} \ar@{->>}[d] & V_{c(134)} \ar@{->>}[dl] \\  &  &  & V_{c(1234)}}

Proof. Some sub-diagrams were already considered in Examples 5.88, 5.108, 5.109. Some remain to be checked, for example, the equality of the composite projections V_{c(12)} ↠ V_{c(123)} ↠ V_{c(1234)} and V_{c(12)} ↠ V_{c(12)(34)} ↠ V_{c(1234)} follows from considering the three CSOs on V_{c(12)} as in Example 5.88. The corresponding composites of inclusions are also equal.

Lemma 5.111. Given V with commuting CSOs J_{V}^{1} , J_{V}^{2} , J_{V}^{3} , J_{V}^{4} , and U with commuting CSOs J_{U}^{1}, J_{U}^{2}, J_{U}^{3}, J_{U}^{4}, if H : U → V satisfies H ∘ J_{U}^{1} = J_{V}^{1} ∘ H and H ∘ J_{U}^{2} = J_{V}^{2} ∘ H and H ∘ J_{U}^{3} = J_{V}^{3} ∘ H and H ∘ J_{U}^{4} = J_{V}^{4} ∘ H, then H respects the corresponding direct sums from Examples 5.108 and 5.109, and induces maps U_{c(i_{1}i_{2})} → V_{c(i_{1}i_{2})}, U_{c(i_{1}i_{2}i_{3})} → V_{c(i_{1}i_{2}i_{3})}, and U_{c(i_{1}i_{2})(i_{3}i_{4})} → V_{c(i_{1}i_{2})(i_{3}i_{4})}, which are c-linear with respect to all pairs of CSOs induced by J_{U}^{i}, J_{V}^{i}, and invertible if H is. The induced map U_{c(1234)} → V_{c(1234)} is c-linear, and invertible if H is.

Proof. All the claims follow from Lemma 5.57 and Lemma 5.90. As in Lemma 5.90, the map U_{c(1234)} → V_{c(1234)} is canonically induced, not depending on the ordering of the indices.

Example 5.112. For V with commuting CSOs J_{V}, J′_{V}, and U with commuting CSOs J_{U}, J′_{U}, the space U ⊗ V has four commuting CSOs:

J_{1} = [J_{U} ⊗ Id_{V}], J_{2} = [Id_{U} ⊗ J_{V}], J_{3} = [J′_{U} ⊗ Id_{V}], J_{4} = [Id_{U} ⊗ J′_{V}].

Theorem 5.110 applies, to give a collection of subspaces of U ⊗ V. From Example 5.64, (U ⊗ V)_{c(13)} = U_{c} ⊗ V and (U ⊗ V)_{c(24)} = U ⊗ V_{c}. From Example 5.65, (U ⊗ V)_{c(12)} = U ⊗_{c} V, and we (temporarily) denote a similar construction (U ⊗ V)_{c(34)} = U ⊗′ V. The subspaces (U ⊗ V)_{c(14)} and (U ⊗ V)_{c(23)} did not appear in previous Examples, and are omitted from the following commutative diagram, where the positions of the objects match the corresponding positions in the diagram from Theorem 5.110.

\xymatrix{ &  &  & U ⊗ V \ar@{->>}[dll] \ar@{->>}[ddlll] \ar@{->>}[drr] \ar@{->>}[ddrrr] \\  & U_{c} ⊗ V \ar@{->>}[ddl] \ar@{->>}[ddddrrr] \ar@{->>}[dddrrrr] &  &  &  & U ⊗ V_{c} \ar@{->>}[ddddlll] \ar@{->>}[ddd] \ar@{->>}[ddr] \\ U ⊗_{c} V \ar@{->>}[d] \ar@{->>}[ddr] \ar@{->>}[dddrr] &  &  &  &  &  & U ⊗′ V \ar@{->>}[ddlllll] \ar@{->>}[dddll] \ar@{->>}[d] \\ U_{c} ⊗_{c} V \ar@{->>}[dddrrr] &  &  &  &  &  & U ⊗′ V_{c} \ar@{->>}[dddlll] \\  & U ⊗′_{c} V \ar@{->>}[ddrr] &  &  &  & U_{c} ⊗ V_{c} \ar@{->>}[ddll] \\  &  & U ⊗_{c} V_{c} \ar@{->>}[dr] &  & U_{c} ⊗′ V \ar@{->>}[dl] \\  &  &  & U_{c} ⊗_{c} V_{c}}

The subspaces U_{c} ⊗_{c} V, U ⊗_{c} V_{c}, U_{c} ⊗′ V, and U ⊗′ V_{c} are as in Example 5.94. The set U ⊗′_{c}V corresponds to (U ⊗V)_{c(12)(34)}, where J_{1} = J_{2} and J_{3} = J_{4}; this notation resembles (5.6) from Example 5.61, with the commuting involutions −J_{1} ∘ J_{2} and −J_{3} ∘ J_{4}.

Example 5.113. For V with commuting CSOs J_{V}, J′_{V}, and U with commuting CSOs J_{U}, J′_{U}, the space Hom(U, V) has four commuting CSOs:

J_{1} = Hom(J_{U}, Id_{V}), J_{2} = Hom(J′_{U}, Id_{V}),
J_{3} = Hom(Id_{U}, J_{V}), J_{4} = Hom(Id_{U}, J′_{V}).

Theorem 5.110 applies, to give a collection of subspaces of Hom(U, V). The subspace (Hom(U, V))_{c(12)} was considered in Example 5.95. As in Example 5.69, denote (Hom(U, V))_{c(13)} = Hom_{c}(U, V), and (temporarily) denote a similar construction (Hom(U, V))_{c(24)} = Hom′(U, V). From Example 5.74, denote (Hom(U, V))_{c(34)} = Hom(U, V_{c}). The subspaces (Hom(U, V))_{c(14)} and (Hom(U, V))_{c(23)} are omitted from the following commutative diagram, but otherwise the positions of the objects match the corresponding positions in the diagram from Theorem 5.110 and Example 5.112.

\xymatrix{ &  &  & Hom(U, V) \ar@{->>}[dll] \ar@{->>}[ddlll] \ar@{->>}[drr] \ar@{->>}[ddrrr] \\  & Hom_{c}(U, V) \ar@{->>}[ddl] \ar@{->>}[ddddrrr] \ar@{->>}[dddrrrr] &  &  &  & Hom′(U, V) \ar@{->>}[ddddlll] \ar@{->>}[ddd] \ar@{->>}[ddr] \\ (Hom(U, V))_{c(12)} \ar@{->>}[d] \ar@{->>}[ddr] \ar@{->>}[dddrr] &  &  &  &  &  & Hom(U, V_{c}) \ar@{->>}[ddlllll] \ar@{->>}[dddll] \ar@{->>}[d] \\ (Hom(U, V))_{c(123)} \ar@{->>}[dddrrr] &  &  &  &  &  & Hom′(U, V_{c}) \ar@{->>}[dddlll] \\  & (Hom(U, V))_{c(12)(34)} \ar@{->>}[ddrr] &  &  &  & Hom′_{c}(U, V) \ar@{->>}[ddll] \\  &  & (Hom(U, V))_{c(124)} \ar@{->>}[dr] &  & Hom_{c}(U, V_{c}) \ar@{->>}[dl] \\  &  &  & (Hom(U, V))_{c(1234)}}

Adapting the notation from (5.6) in Example 5.61, Hom′_{c}(U, V) denotes the subspace

(Hom(U, V))_{c(13)(24)} = {A : U → V : A ∘ J_{U} = J_{V} ∘ A and A ∘ J′_{U} = J′_{V} ∘ A}.

If we ignore J′_{U}, then the two projections onto Hom_{c}(U, V_{c}) in the above diagram are as in Example 5.98. Similarly ignoring J_{U}, the two projections onto Hom′(U, V_{c}) are also as in Example 5.98.

Example 5.114. The space (Hom(U, V))_{c(12)} from Example 5.113 is related to Hom(U_{c}, V) as in Example 5.95, by an invertible map P_{c(12)} ∘ Hom(P_{c}, Id_{V}). Both (Hom(U, V))_{c(12)} and Hom(U_{c}, V) admit three induced CSOs, and Hom(U_{c}, V) admits three direct sums as in Example 5.98. The map P_{c(12)} ∘ Hom(P_{c}, Id_{V}) is c-linear with respect to the three corresponding pairs of CSOs, so by Lemma 5.90, it respects the direct sums and induces c-linear invertible maps as indicated by the unlabeled horizontal arrows in the following diagram. The left part is copied from the diagram in Example 5.113, and the top triangle and top square appeared already in the diagram for Example 5.95.

\xymatrix{ &  &  & Hom(U, V) \ar@{->>}[dlll]^{P_{c(12)}} \ar[d]^{Hom(Q_{c},Id_{V} )} \\ (Hom(U, V))_{c(12)} \ar@{->>}[d] \ar@{->>}[ddr] \ar@{->>}[dddrr] \ar[rrr]_{Hom(Q_{c},Id_{V} )∘Q_{c(12)}} \ar@{^{(}->}[urrr]^{Q_{c(12)}} &  &  & Hom(U_{c}, V) \ar@{->>}[d] \ar@{->>}[ddr] \ar@{->>}[dddrr] \ar[lll]_{P_{c(12)}∘Hom(P_{c},Id_{V} )} \ar[u]^{Hom(P_{c},Id_{V} )} \\ (Hom(U, V))_{c(123)} \ar@{->>}[dddrrr] \ar[rrr] &  &  & Hom_{c}(U_{c}, V) \ar@{->>}[dddrr]_{P^{2}} \ar[lll] \\  & (Hom(U, V))_{c(12)(34)} \ar@{->>}[ddrr] \ar[rrr] &  &  & Hom(U_{c}, V_{c}) \ar@{->>}[ddr]^{P^{1}} \ar[lll] \\  &  & (Hom(U, V))_{c(124)} \ar@{->>}[dr] \ar[rrr] &  &  & Hom′(U_{c}, V) \ar@{->>}[d] \ar[lll] \\  &  &  & (Hom(U, V))_{c(1234)} \ar[rr] &  & Hom_{c}(U_{c}, V_{c}) \ar[ll]}

The projections P^{1}, P^{2} are labeled to match (5.14), (5.16) from Example 5.98, and the lower right vertical arrow is also from (5.16).

Theorem 5.115. For V with commuting CSOs J_{V}, J′_{V}, and U with commuting CSOs J_{U}, J′_{U}, let W be a space with four commuting CSOs J_{W}^{1} , J_{W}^{2} , J_{W}^{3} J_{W}^{4} . If H : W → Hom(U, V) satisfies Hom(J_{U}, Id_{V})∘H = H∘J_{W}^{1} and Hom(J′_{U}, Id_{V})∘H = H ∘ J_{W}^{2} and Hom(Id_{U}, J_{V}) ∘ H = H ∘ J_{W}^{3} and Hom(Id_{U}, J′_{V}) ∘ H = H ∘ J_{W}^{4} , then H respects the corresponding direct sums from Examples 5.113 and 5.114, and the induced maps are c-linear. If also H is invertible, then the induced maps are invertible.

Proof. The claims for the induced maps

W_{c(i_{1}i_{2})} → (Hom(U, V))_{c(i_{1}i_{2})}
W_{c(i_{1}i_{2}i_{3})} → (Hom(U, V))_{c(i_{1}i_{2}i_{3})}
W_{c(i_{1}i_{2})(i_{3}i_{4})} → (Hom(U, V))_{c(i_{1}i_{2})(i_{3}i_{4})}
ã_{3} : W_{c(1234)} → (Hom(U, V))_{c(1234)}

follow from Lemma 5.111. The target spaces are as in the diagram from Example 5.113, for example, H induces a map W_{c(13)(24)} → Hom′_{c}(U, V), labeled a_{3} in the diagram below, and a_{3} induces ã_{3}. The claims for the induced maps

a_{2} = Hom(Q_{c}, Id_{V}) ∘ H ∘ Q′_{c(12)} : W_{c(12)} → Hom(U_{c}, V),

and W_{c(123)} → Hom_{c}(U_{c}, (V, J_{V})) (= Hom_{c}(U_{c}, V) in the diagram from Example 5.114), and W_{c(124)} → Hom_{c}(U_{c}, (V, J′_{V})) = Hom′(U_{c}, V) follow from Lemma 5.96. The map a_{2} is c-linear with respect to the pair of CSOs induced by J_{W}^{3} and Hom(Id_{U}, J_{V}), as mentioned in the Proof of Lemma 5.96, and is also c-linear with respect to the pair of CSOs induced by J_{W}^{4} and Hom(Id_{U}, J′_{V}), so it satisfies the hypotheses of Lemma 5.90, and respects the corresponding direct sums in the diagram from Example 5.114. The maps induced by a_{2} are c-linear:

W_{c(12)(34)} → Hom(U_{c}, V_{c})
ã_{2} : W_{c(1234)} → Hom_{c}(U_{c}, V_{c}).

In the following diagram, the left half is copied from the diagram from Example 5.114, where a_{1} = Hom(Q_{c}, Id_{V}) ∘ Q_{c(12)} induces ã_{1}, and they are both c-linear and invertible. The space Hom′_{c}(U, V) and projections P_{c(13)(24)}, P_{3} are copied from Example 5.113, and the right half of the diagram is part of the diagram from Theorem 5.110.

[figure: Hom(U, V); P_{c(12)}; Hom(Q_{c},Id_{V} ); P_{c(13)(24)}; W; P′_{c(12)}; P′_{c(13)(24)}; H; (Hom(U, V))_{c(12)}; a_{1}; P_{1}; Hom(U_{c}, V); P_{2}; W_{c(12)}; a_{2}; P′_{1}; Hom′_{c}(U, V); P_{3}; W_{c(13)(24)}; a_{3}; P′_{2}; W_{c(1234)}; ã_{3}; ã_{2}; (Hom(U, V))_{c(1234)}; ã_{1}; Hom_{c}(U_{c}, V_{c})]

Finally, we remark that ã_{2} = ã_{1} ∘ ã_{3}; an analogous property was observed in the Proof of Lemma 5.96. The identity can be checked directly, using the c-linearity of H. The map ã_{2} acts on w ∈ W_{c(1234)} as: ã_{2} : w ↦ P′_{c} ∘ H(w) ∘ Q_{c}, where P′_{c} is the projection V ↠ V_{c}.

Exercise 5.116. Given U, V, W, with U = (U, J_{U}), V = (V, J_{V}), and W = (W, J), for any A ∈ Hom_{c}(U, V), the map t_{UV}^{W} (A) : Hom(V, W) → Hom(U, W) (or, more precisely, t_{UV}^{W} (Q′_{c}(A)), where Q′_{c} is the inclusion of Hom_{c}(U, V) in Hom(U, V)) is c-linear with respect to both pairs Hom(Id_{V} , J_{W}), Hom(Id_{U}, J_{W}) and Hom(J_{U}, Id_{W}), Hom(J_{V} , Id_{W}), so t_{UV}^{W} (A) respects the direct sums and induces a c-linear map Hom_{c}(V, W) → Hom_{c}(U, W). The resulting map, denoted

t_{UV}^{W} : Hom_{c}(U, V) → Hom_{c}(Hom_{c}(V, W), Hom_{c}(U, W)),

is c-linear.

Hint. Lemma 5.57, Exercise 5.80, and Exercise 5.38 apply. The last claim can be checked directly. However, by following the diagrams from Examples 5.113 and 5.114, a little more can be obtained. Let Q_{i} and P_{i} denote the operators for the direct sum Hom(V, W) = Hom_{c}(V, W) ⊕ Hom_{a}(V, W), so that

Hom(Q_{c}, Id_{Hom(U,W)}) :
Hom(Hom(V, W), Hom(U, W)) → Hom(Hom_{c}(V, W), Hom(U, W))

is as in Example 5.114. Then

Hom(Q_{c}, Id_{Hom(U,W)}) ∘ t_{UV}^{W} : Hom(U, V) → Hom(Hom_{c}(V, W), Hom(U, W))

is c-linear with respect to Hom(J_{U}, Id_{V}) and the CSO induced by

J^{3} = Hom(Id_{Hom(V,W)}, Hom(J_{U}, Id_{W})),

and is also c-linear with respect to Hom(Id_{U}, J_{V}) and the CSO induced by

J^{1} = Hom(Hom(J_{V} , Id_{W}), Id_{Hom(U,W)}).

So by Lemma 5.57, it induces a c-linear map

Hom_{c}(U, V) → Hom_{c}(Hom_{c}(V, W), Hom(U, W)).

As claimed above, the image of this induced map is contained in the subspace Hom_{c}(Hom_{c}(V, W), Hom_{c}(U, W)). For A ∈ Hom_{c}(U, V) and K ∈ Hom_{c}(V, W),

(t_{UV}^{W} (Q′_{c}(A))) ∘ Q_{c} : K ↦ (Q_{c}(K)) ∘ (Q′_{c}(A)) = K ∘ A ∈ Hom_{c}(U, W).

Example 5.117. Given U_{1}, U_{2}, V_{1}, V_{2}, and U_{1} = (U_{1}, J_{U_{1}}), the canonical map (Definition 1.33)

j : Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) → Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})

is c-linear with respect to the induced CSOs, by Lemma 1.37. A similar statement holds if any one of the four spaces has a CSO.

If every one of the above four spaces has a CSO, U_{1} = (U_{1}, J_{U_{1}}), U_{2} = (U_{2}, J_{U_{2}}), V_{1} = (V_{1}, J_{V_{1}}), V_{2} = (V_{2}, J_{V_{2}}), then Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) admits four commuting CSOs:

J′_{1} = [Hom(J_{U_{1}}, Id_{V_{1}}) ⊗ Id_{Hom(U_{2},V_{2})}]
J′_{2} = [Id_{Hom(U_{1},V_{1})} ⊗ Hom(J_{U_{2}}, Id_{V_{2}})]
J′_{3} = [Hom(Id_{U_{1}}, J_{V_{1}}) ⊗ Id_{Hom(U_{2},V_{2})}]
J′_{4} = [Id_{Hom(U_{1},V_{1})} ⊗ Hom(Id_{U_{2}}, J_{V_{2}})],

and Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) also admits four commuting CSOs:

J_{1} = Hom([J_{U_{1}} ⊗ Id_{U_{2}}], Id_{V_{1}⊗V_{2}})
J_{2} = Hom([Id_{U_{1}} ⊗ J_{U_{2}}], Id_{V_{1}⊗V_{2}})
J_{3} = Hom(Id_{U_{1}⊗U_{2}}, [J_{V_{1}} ⊗ Id_{V_{2}}])
J_{4} = Hom(Id_{U_{1}⊗U_{2}}, [Id_{V_{1}} ⊗ J_{V_{2}}]).

Since j is c-linear with respect to each pair J′_{i}, J_{i}, Theorem 5.115 applies, with H = j and W = Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}), so j induces maps on corresponding subspaces, which are c-linear with respect to (possibly several pairs of) corresponding CSOs, and which are invertible if j is. From the diagrams in Theorem 5.110 and Examples 5.112, 5.113, some induced maps from Lemma 5.111 are evident:

Hom(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})
Hom(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})
Hom_{c}(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})
Hom(U_{1}, V_{1}) ⊗ Hom_{c}(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})
Hom(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → Hom(U_{1} ⊗ U_{2}, V_{1} ⊗_{c} V_{2})
Hom(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗_{c} V_{2})
(5.22) a_{3} : Hom_{c}(U_{1}, V_{1}) ⊗ Hom_{c}(U_{2}, V_{2}) → Hom′_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2})
Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗_{c} V_{2})
ã_{3} : Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) → (Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}))_{c(1234)}.

For example, the seventh map, labeled a_{3} as in the diagram from Theorem 5.115, is c-linear with respect to both corresponding pairs of induced CSOs, and for c-linear maps A and B, takes A ⊗ B to the map j(A ⊗ B) : U_{1} ⊗ U_{2} → V_{1} ⊗ V_{2}, which is c-linear with respect to the pair [J_{U_{1}} ⊗ Id_{U_{2}}], [J_{V_{1}} ⊗ Id_{V_{2}}], and also c-linear with respect to the pair [Id_{U_{1}} ⊗ J_{U_{2}}], [Id_{V_{1}} ⊗ J_{V_{2}}].

Also, if some but not all of the four spaces have CSOs, then there may still be some induced maps, for example, the first one in the above list makes sense if only U_{1} and V_{2} have CSOs.

Let Q′_{c(12)}, P′_{c(12)}, and Q_{c(12)}, P_{c(12)} denote the inclusion and projection operators for the direct sums produced by J′_{1}, J′_{2}, and J_{1}, J_{2}, respectively, as appearing in the diagram from the Proof of Theorem 5.115. By Lemma 5.76, the map j also respects the direct sum

Hom(U_{1} ⊗_{c} U_{2}, V_{1} ⊗ V_{2}) ⊕ Hom(U_{1} ⊗_{a} U_{2}, V_{1} ⊗ V_{2}),

and induces a c-linear map, labeled

a_{2} = Hom(Q_{c}, Id_{V_{1}⊗V_{2}}) ∘ j ∘ Q′_{c(12)},

as in the following diagram, a copy of two blocks of the diagram from Theorem 5.115.

\xymatrix{Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) \ar[d]_{Hom(Q_{c},Id_{V_{1}⊗V_{2}} )} & Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) \ar[l]_{j} \ar@{->>}[d]^{P′_{c(12)}} \\ Hom(U_{1} ⊗_{c} U_{2}, V_{1} ⊗ V_{2}) \ar@{->>}[d]_{P_{2}} & Hom(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) \ar[l]_{a_{2}} \ar@{->>}[d]^{P′_{1}} \\ Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗_{c} V_{2}) & Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) \ar[l]_{ã_{2}}}

The map a_{2} is equal to the composite of the induced map

P_{c(12)} ∘ j ∘ Q′_{c(12)} : Hom(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → (Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}))_{c(12)}

with the invertible c-linear map from Example 5.114, labeled a_{1} in Theorem 5.115,

Hom(Q_{c}, Id_{V_{1}⊗V_{2}})∘Q_{c(12)} : (Hom(U_{1}⊗U_{2}, V_{1}⊗V_{2}))_{c(12)} → Hom(U_{1}⊗_{c}U_{2}, V_{1}⊗V_{2}).

Also, a_{2} is c-linear with respect to all three corresponding pairs of induced CSOs, so as in Example 5.113 and Theorem 5.115, it respects the corresponding direct sums, to induce c-linear maps:

Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗ V_{2})
Hom(U_{1}, V_{1}) ⊗′_{c} Hom(U_{2}, V_{2}) → Hom(U_{1} ⊗_{c} U_{2}, V_{1} ⊗_{c} V_{2})
Hom(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗ V_{2})
ã_{2} : Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) → Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗_{c} V_{2}).

As remarked in the Proof of Theorem 5.115, let ã_{1} denote the invertible map induced by a_{1},

(Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}))_{c(1234)} → Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗_{c} V_{2}),

so that then ã_{2} = ã_{1} ∘ ã_{3}. The map ã_{2} is invertible if j is; it acts on w ∈ Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) ⊆ Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}) as: ã_{2} : w ↦ P′_{c} ∘ j(w) ∘ Q_{c}, where P′_{c} is the projection V_{1} ⊗ V_{2} ↠ V_{1} ⊗_{c} V_{2}.

Example 5.118. For U_{1}, U_{2}, V_{1}, V_{2} as in Example 5.117, suppose A : U_{1} → V_{1} and B : U_{2} → V_{2} are c-linear. Then

A ⊗ B ∈ Hom_{c}(U_{1}, V_{1}) ⊗ Hom_{c}(U_{2}, V_{2}) ⊆ Hom(U_{1}, V_{1}) ⊗ Hom(U_{2}, V_{2}),

so A ⊗ B is in the domain of a_{3} = j|_{Hom_{c}(U_{1},V_{1})⊗Hom_{c}(U_{2},V_{2})} as in (5.22) and the following diagram, a copy of two blocks of the diagram from Theorem 5.115.

\xymatrix{Hom′_{c}(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) \ar@{->>}[dd]_{P_{3}} & Hom_{c}(U_{1}, V_{1}) ⊗ Hom_{c}(U_{2}, V_{2}) \ar[l]_{a_{3}} \ar@{->>}[d]^{P′_{2}} \\  & Hom_{c}(U_{1}, V_{1}) ⊗_{c} Hom_{c}(U_{2}, V_{2}) \ar[d]^{ã_{2}} \ar@{-->}[dl]_{ã_{3}} \\ (Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}))_{c(1234)} \ar[r]_{ã_{1}} & Hom_{c}(U_{1} ⊗_{c} U_{2}, V_{1} ⊗_{c} V_{2})}

Starting with A ⊗ B, these outputs are equal by the commutativity of the diagram from Theorem 5.115:

(ã_{2} ∘ P′_{2})(A ⊗ B) = ã_{2}(\frac{1}{2}(A ⊗ B − (J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})))

(5.23) = P′_{c} ∘ (\frac{1}{2}([A ⊗ B] − [(J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})])) ∘ Q_{c},

(ã_{1} ∘ P_{3} ∘ a_{3})(A ⊗ B) = ã_{1}(P_{3}([A ⊗ B]))

= P′_{c} ∘ (\frac{1}{2}([A ⊗ B] − [J_{V_{1}} ⊗ Id_{V_{2}}] ∘ [A ⊗ B] ∘ [Id_{U_{1}} ⊗ J_{U_{2}}])) ∘ Q_{c}.

As in Lemma 5.71, this is the restriction of the map \frac{1}{2}([A ⊗ B] − [(J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})]) ∈ Hom(U_{1} ⊗ U_{2}, V_{1} ⊗ V_{2}) to the subspace U_{1} ⊗_{c} U_{2} in the domain and V_{1} ⊗_{c} V_{2} in the target.

For u_{1} ∈ U_{1}, u_{2} ∈ U_{2}, these elements of V_{1} ⊗ V_{2} are equal:

([A ⊗ B] ∘ Q_{c})(P_{c}(u_{1} ⊗ u_{2}))
= [A ⊗ B](\frac{1}{2}(u_{1} ⊗ u_{2} − (J_{U_{1}}(u_{1})) ⊗ (J_{U_{2}}(u_{2}))))
= \frac{1}{2}((A(u_{1})) ⊗ (B(u_{2})) − (A(J_{U_{1}}(u_{1}))) ⊗ (B(J_{U_{2}}(u_{2})))),
(−[(J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})] ∘ Q_{c})(P_{c}(u_{1} ⊗ u_{2}))
= −[(J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})](\frac{1}{2}(u_{1} ⊗ u_{2} − (J_{U_{1}}(u_{1})) ⊗ (J_{U_{2}}(u_{2}))))
= \frac{1}{2}(−(J_{V_{1}}(A(u_{1}))) ⊗ (B(J_{U_{2}}(u_{2}))) + (J_{V_{1}}(A(J_{U_{1}}(u_{1})))) ⊗ (B(J_{U_{2}}(J_{U_{2}}(u_{2}))))).

Because U_{1} ⊗_{c} U_{2} is spanned by elements of the form P_{c}(u_{1} ⊗ u_{2}), these maps U_{1} ⊗_{c} U_{2} → V_{1} ⊗_{c} V_{2} are equal:

P′_{c} ∘ (\frac{1}{2}([A ⊗ B] − [(J_{V_{1}} ∘ A) ⊗ (B ∘ J_{U_{2}})])) ∘ Q_{c} = P′_{c} ∘ [A ⊗ B] ∘ Q_{c}.

The above LHS is as in (5.23), and the RHS is from Notation 5.67, giving the equality:

(ã_{2} ∘ P′_{2})(A ⊗ B) = [A ⊗_{c} B] = (Hom(Q_{c}, P′_{c}) ∘ j|_{Hom_{c}(U_{1},V_{1})⊗Hom_{c}(U_{2},V_{2})})(A ⊗ B).

There would be no ambiguity in denoting j = ã_{2}.

Lemma 5.119. For any U, V, W, the following diagram is commutative.

\xymatrix{U_{c} ⊗_{c} Hom_{c}(V, W) ⊗_{c} V \ar[d]_{[n⊗_{c}Id_{V} ]} \ar[r]^{[Id_{U} ⊗_{c}Ev_{VW}^{c}]} & U ⊗_{c} W \\ Hom_{c}(V, U ⊗_{c} W) ⊗_{c} V \ar[ur]_{Ev_{V,U⊗cW}^{c}}}

Proof. This is a c-linear version of Lemma 2.76, so all these objects are subspaces of spaces from Lemma 2.76; the following argument keeps track of all of the inclusions. The spaces U ⊗ Hom(V, W) ⊗ V and Hom(V, U ⊗ W) ⊗ V each admit four commuting CSOs, and the map [n ⊗ Id_{V}] is c-linear with respect to four corresponding pairs as in Lemma 5.111, by Lemma 5.24 and Theorem 5.100. Some of the projections and induced maps from Lemma 5.111 are as in the following diagram.

\xymatrix{U ⊗ Hom(V, W) ⊗ V \ar[r]^{[n⊗Id_{V} ]} \ar@{->>}[d]_{P_{c(23)}} & Hom(V, U ⊗ W) ⊗ V \ar@{->>}[d]^{P′_{c(23)}} \\ U ⊗ Hom_{c}(V, W) ⊗ V \ar@{->>}[d]_{P_{c((23)1)}} \ar[r]^{[n^{1}⊗Id_{V} ]} & Hom_{c}(V, U ⊗ W) ⊗ V \ar@{->>}[d]^{P′_{c((23)1)}} \\ U ⊗_{c} Hom_{c}(V, W) ⊗ V \ar[r]^{[n⊗Id_{V} ]} \ar@{->>}[d]_{P_{c((231)4)}} & Hom_{c}(V, U ⊗_{c} W) ⊗ V \ar@{->>}[d]^{P′_{c((231)4)}} \\ U ⊗_{c} Hom_{c}(V, W) ⊗_{c} V \ar[r]^{[n⊗_{c}Id_{V} ]} & Hom_{c}(V, U ⊗_{c} W) ⊗_{c} V}

In the upper block, [n^{1} ⊗ Id_{V}] is induced by [n ⊗ Id_{V}] as in Lemma 1.87 and Theorem 5.100, where n induces n^{1}. The middle block is similarly related to the lower block from Theorem 5.100. In the lowest block, [n ⊗_{c} Id_{V}] is induced as in Example 5.66, and [n ⊗_{c} Id_{V}] also appears in the following diagram.

\xymatrix{U ⊗ Hom(V, W) ⊗ V \ar[ddrr]^{[Id_{U} ⊗Ev_{V W}]} \\ U ⊗ Hom_{c}(V, W) ⊗ V \ar@{^{(}->}[u]^{Q_{c(23)}} \\ U ⊗ Hom_{c}(V, W) ⊗_{c} V \ar@{^{(}->}[u]^{Q_{c((23)4)}} \ar[rr]^{[Id_{U} ⊗Ev_{VW}^{c}]} &  & U ⊗ W \ar@{->>}[dl]_{P′_{c}} \\ U ⊗_{c} Hom_{c}(V, W) ⊗_{c} V \ar[d]_{[n⊗_{c}Id_{V} ]} \ar[r]^{[Id_{U} ⊗_{c}Ev_{VW}^{c}]} \ar@{^{(}->}[u]^{Q_{c((234)1)}} & U ⊗_{c} W \\ Hom_{c}(V, U ⊗_{c} W) ⊗_{c} V \ar[ur]_{Ev_{V,U⊗cW}^{c}} \ar@{^{(}->}[d]_{Q′_{c((132)4)}} \\ Hom_{c}(V, U ⊗_{c} W) ⊗ V \ar@{^{(}->}[d]_{Q′_{c((13)2)}} \\ Hom(V, U ⊗_{c} W) ⊗ V \ar[uuur]^{Ev_{V,U⊗cW}} \\ Hom(V, U ⊗ W) ⊗ V \ar@{->>}[u]^{P′_{c(13)}} \ar[uuuuurr]_{Ev_{V,U⊗W}}}

In the upper block, Q_{c(23)} is the inclusion corresponding to the projection P_{c(23)} from the previous diagram, and as in Example 5.64, Q_{c(23)} = [Id_{U} ⊗ [Q_{c} ⊗ Id_{V}]] for Q_{c} : Hom(V, W) ↪ Hom(V, W). The inclusion Q_{c((23)4)} does not correspond to any projection from the previous diagram, but again as in Example 5.64, it is equal to [Id_{U} ⊗ Q̃], for the inclusion Q̃ : Hom_{c}(V, W) ⊗_{c} V ↪ Hom_{c}(V, W) ⊗ V. The upper block is then commutative by Lemma 1.36 and adapting formula (5.20) from Example 5.106, Ev_{VW}^{c} = Ev_{V W} ∘ [Q_{c} ⊗ Id_{V}] ∘ Q̃. The second block is commutative by Notation 5.67 for the induced map [Id_{U} ⊗_{c} Ev_{VW}^{c}]. The commutativity of the center block is the claim of the Lemma. The fourth block is commutative, it is an adaptation of formula (5.20) defining Ev_{V,U⊗_{c}W}^{c}. The inclusion Q′_{c((132)4)} is equal to the inclusion with the first three indices re-ordered, Q′_{c((231)4)}, corresponding to the projection in the first diagram. In the lowest block, P′_{c(13)} does not appear in the first diagram; it is equal to [Hom(Id_{V} , P′_{c}) ⊗ Id_{V}] as in Example 5.64 and Example 5.74. So, the lowest block is commutative by Lemma 2.72. The claim follows, using Lemma 2.76 and equalities of composite inclusions from Theorem 5.110.

Ev_{V,U⊗_{c}W}^{c} ∘ [n ⊗_{c} Id_{V}]
= Ev_{V,U⊗_{c}W} ∘ Q′_{c((13)2)} ∘ Q′_{c((132)4)} ∘ [n ⊗_{c} Id_{V}]
= Ev_{V,U⊗_{c}W} ∘ P′_{c(13)} ∘ Q′_{c(13)} ∘ Q′_{c((13)2)} ∘ Q′_{c((132)4)} ∘ [n ⊗_{c} Id_{V}]
= P′_{c} ∘ Ev_{V,U⊗W} ∘ Q′_{c(23)} ∘ Q′_{c((23)1)} ∘ Q′_{c((231)4)} ∘ [n ⊗_{c} Id_{V}]
= P′_{c} ∘ Ev_{V,U⊗W} ∘ [n ⊗ Id_{V}] ∘ Q_{c(23)} ∘ Q_{c((23)1)} ∘ Q_{c((231)4)}
= P′_{c} ∘ [Id_{U} ⊗ Ev_{V W}] ∘ Q_{c(23)} ∘ Q_{c((23)4)} ∘ Q_{c((234)1)}
= [Id_{U} ⊗_{c} Ev_{VW}^{c}].

Theorem 5.120. For any V = (V, J_{V}), U, W, and c-linear F : V ⊗_{c} U → V ⊗_{c} W, if V is finite-dimensional then the n maps in the following diagram are invertible:

\xymatrix{V ⊗_{c} U \ar[r]^{F} & V ⊗_{c} W \\ V ⊗_{c} Hom_{c}(V, U) ⊗_{c} V \ar[d]_{[n_{2}⊗_{c}Id_{V} ]} \ar[u]^{[Id_{V} ⊗_{c}Ev_{VU}^{c}]} & V ⊗_{c} Hom(V, W) ⊗_{c} V \ar[d]^{[n′_{2}⊗_{c}Id_{V} ]} \ar[u]_{[Id_{V} ⊗_{c}Ev_{VW}^{c}]} \\ Hom(V, V ⊗_{c} U) ⊗_{c} V \ar[r]_{[Hom_{c}(Id_{V} ,F)⊗_{c}Id_{V} ]} & Hom(V, V ⊗_{c} W) ⊗_{c} V}

and the diagram is commutative, in the sense that

F ∘ [Id_{V} ⊗_{c} Ev_{VU}^{c}] ∘ [n_{2} ⊗_{c} Id_{V}]^{−1}
= [Id_{V} ⊗_{c} Ev_{VW}^{c}] ∘ [n′_{2} ⊗_{c} Id_{V}]^{−1} ∘ [Hom_{c}(Id_{V} , F) ⊗_{c} Id_{V}].

Proof. This is a c-linear version of Theorem 2.77, and the Proof is analogous. The maps n_{2} and n′_{2} are special cases of the n map from Lemma 5.119; they are invertible by Lemma 1.43 and Theorem 5.100, with [n_{2} ⊗_{c} Id_{V}]^{−1} = [n_{2}^{−1} ⊗_{c} Id_{V}].

By Lemma 5.119, the upward composite on the left is equal to Ev_{V,V⊗_{c}U}^{c}, and similarly the upward composite on the right is equal to Ev_{V,V⊗_{c}W}^{c}. The claim then follows from Lemma 5.107.

Remark 5.121. The results in this Section on the c-linear evaluation map from Example 5.106: Lemma 5.107, Lemma 5.119, and Theorem 5.120, give some details omitted from [C_{2}] §4.

5.4. Real trace with complex vector values

In this Section we develop the notion of vector valued trace of ℝ-linear maps, where the value spaces have complex structure operators. The approach will be to refer to Chapter 2, while avoiding scalar multiplication.

Theorem 5.100 on the c-linearity of n maps generalizes in a straightforward way to the various orderings of n maps from Notation 1.40, as in the following Corollary. Recall from Theorem 2.73 the special case

n′ : Hom(V, W) ⊗ V → Hom(V, V ⊗ W) : A ⊗ v ↦ (u ↦ v ⊗ (A(u))),

which is invertible if V is finite-dimensional.

Corollary 5.122. If V = (V, J_{V}) and W = (W, J_{W}), then n′ : Hom(V, W) ⊗ V → Hom(V, V ⊗ W) is c-linear with respect to corresponding pairs of the three commuting CSOs induced on each space, so it respects the direct sums and induces maps

n′_{1} : Hom_{c}(V, W) ⊗ V → Hom_{c}(V, V ⊗ W)
n′_{2} : Hom(V, W) ⊗_{c} V → Hom_{c}(V, V ⊗ W)
n′_{3} : Hom(V, W) ⊗_{c} V → Hom(V, V ⊗_{c} W)
n′ : Hom_{c}(V, W) ⊗_{c} V → Hom_{c}(V, V ⊗_{c} W),

which are invertible if n′ is.

Theorem 5.123. If V is finite-dimensional and W = (W, J_{W}), then the map

T r_{V;W} = Ev_{V W} ∘ (n′)^{−1} : Hom(V, V ⊗ W) → W

is c-linear.

Proof. The map n′ is from Corollary 5.122: it is c-linear with respect to [Hom(Id_{V} , J_{W}) ⊗ Id_{V}] and Hom(Id_{V} , [Id_{V} ⊗ J_{W}]) (without assuming any CSO on V). The canonical evaluation Ev_{V W} : A⊗v ↦ A(v) from Definition 2.70 is c-linear Hom(V, W) ⊗ V → W as in Exercise 5.44, and the equality T r_{V;W} ∘ n′ = Ev_{V W} is from Theorem 2.73. The result could also be proved by applying Corollary 2.58 (or Corollary 2.74) with B = J_{W}.

Theorem 5.124. If V is finite-dimensional and W admits commuting CSOs J_{1}, J_{2}, then the map T r_{V;W} respects the direct sums

Hom(V, V ⊗ W_{c}) ⊕ Hom(V, V ⊗ W_{a}) → W_{c} ⊕ W_{a},

and the induced c-linear map Hom(V, V ⊗ W_{c}) → W_{c} is equal to T r_{V;W_{c}}.

Proof. Lemma 2.60 applies. The direct sums on V ⊗ W and Hom(V, V ⊗ W) are as in Example 5.64 and Example 5.74, with canonical projections as indicated in the diagram. T r_{V;W} is c-linear with respect to both corresponding pairs of CSOs by Theorem 5.123, and the c-linearity of the induced map follows from Lemma 5.57.

\xymatrix{Hom(V, V ⊗ W) \ar[r]^{T r_{V ;W}} \ar@{->>}[d]_{Hom(Id_{V} ,[Id_{V} ⊗P_{c}])} & W \ar@{->>}[d]^{P_{c}} \\ Hom(V, V ⊗ W_{c}) \ar[r]^{T r_{V ;Wc}} & W_{c}}

Theorem 5.125. For finite-dimensional V, and U, W with CSOs J_{U}, J_{W}, the generalized trace

T r_{V;U,W} : Hom(V ⊗ U, V ⊗ W) → Hom(U, W)

is c-linear with respect to both pairs of corresponding commuting CSOs, and respects the direct sums, inducing a c-linear map, denoted

T r_{V;U,W} : Hom_{c}(V ⊗ U, V ⊗ W) → Hom_{c}(U, W).

Proof. The c-linearity claims follow from Theorem 2.30, and then Lemma 5.57 applies. That is enough for the Proof, but to see how the generalized trace is related to two different vector valued traces, consider the following diagram,

\xymatrix{Hom(V ⊗ U, V ⊗ W) \ar@{-->}[drrr]^{T r_{V ;U,W}} \ar@{->>}[dd] \ar[dr]^{q^{−1}} \\  & M_{11} \ar[ul]^{q} \ar[r]^{Hom(Id_{V} ,n^{−1})} \ar@{->>}[d] & M_{12} \ar[l]^{Hom(Id_{V} ,n)} \ar[r]_{T r_{V ;Hom(U,W)}} \ar@{->>}[d] & Hom(U, W) \ar@{->>}[d]^{P_{H}} \\ Hom_{c}(V ⊗ U, V ⊗ W) \ar[r] & M_{21} \ar[r] \ar[l]^{q_{1}} & M_{22} \ar[l] \ar[r]_{T r_{V ;Homc(U,W)}} & Hom_{c}(U, W)}

where

M_{11} = Hom(V, Hom(U, V ⊗ W))
M_{12} = Hom(V, V ⊗ Hom(U, W))
M_{21} = Hom(V, Hom_{c}(U, V ⊗ W))
M_{22} = Hom(V, V ⊗ Hom_{c}(U, W)).

All the vertical arrows are the canonical projections of the direct sums produced by the commuting CSOs induced by J_{U} and J_{W}. The right square is commutative by Lemma 2.60; this is an example of Theorem 5.124, where the projection M_{12} ↠ M_{22} is equal to Hom(Id_{V} , [Id_{V} ⊗ P_{H}]). The map n : V ⊗ Hom(U, W) → Hom(U, V ⊗ W) from Theorem 5.100 is invertible and c-linear with respect to the commuting corresponding pairs of CSOs induced by J_{U} and J_{W}. Example 5.79 applies to Hom(Id_{V} , n) and the middle square in the diagram: the induced map (lower middle arrow) is invertible, c-linear, and equal to Hom(Id_{V} , n^{1}), where n^{1} is the induced map from Theorem 5.100. The map q is as in Theorem 2.52, which asserts the commutativity of the diagram’s top triangle. By Example 5.101, the map q similarly induces an invertible c-linear map, q_{1}. We can conclude, for K : V → V ⊗ Hom_{c}(U, W),

T r_{V;Hom_{c}(U,W)}(K) = T r_{V;U,W}(q_{1}(n^{1} ∘ K)).

CHAPTER 6 Appendices

6.1. Appendix: Functions and Binary Operations

Definition 6.1. Given sets S and T, the \underline{product} set S ×T is the set of ordered pairs {(σ, τ) : σ ∈ S, τ ∈ T }.

Definition 6.2. Given sets S and T, suppose there is a subset G ⊆ S × T with the following properties.

• If (s_{1}, t_{1}) ∈ G and (s_{2}, t_{2}) ∈ G and s_{1} = s_{2} then t_{1} = t_{2}.

• For each s ∈ S, there is an element (s, t) ∈ G. Then for each s ∈ S, there is exactly one element α(s) ∈ T so that (s, α(s)) ∈ G. This defines a \underline{function} α, with \underline{domain} S, \underline{target} T, and \underline{graph} G, which is denoted (as in Notation 0.36) by the arrow notation α : S ⇝ T.

Definition 6.3. Given a set S, a \underline{binary operation on S} is any function from S × S to S. The notation (S, ∗) denotes a set S, together with ∗, a binary operation on S. For x, y ∈ S, and a binary operation ∗, the element ∗((x, y)) will be abbreviated x ∗ y.

Definition 6.4. A binary operation ∗ on S is \underline{associative} means: for all x, y, z ∈ S, (x ∗ y) ∗ z = x ∗ (y ∗ z). The binary operation ∗ is \underline{commutative} means: for all x, y ∈ S, x ∗ y = y ∗ x.

Definition 6.5. Given (S, ∗), any element e ∈ S such that e ∗ x = x ∗ e = x for all x ∈ S is called an \underline{identity element}.

Exercise 6.6. Given (S, ∗), suppose there is an identity element e ∈ S. Then, the identity element is unique.

Exercise 6.7. Given (S, ∗), with an identity element e, if for all x, y, z ∈ S, x ∗ (y ∗ z) = (x ∗ z) ∗ y, then ∗ is commutative and associative.

Exercise 6.8. Give an example of a set and an operation ∗ where x ∗ (y ∗ z) = (x ∗ z) ∗ y holds but ∗ is not associative.

Definition 6.9. Given (S, ∗), and an identity element e ∈ S, and x, y ∈ S, \underline{y is a ∗-inverse for x} means that x ∗ y = y ∗ x = e.

Note that ∗-inverse cannot be defined without an identity element, so in any statement asserting the existence of a ∗-inverse, it is assumed that there exists an identity element for the operation ∗.

Exercise 6.10. Given (S, ∗), let e be an identity element. Then e has a ∗- inverse, and this inverse is unique.

Exercise 6.11. Given (S, ∗), and x ∈ S, if ∗ is associative, and there exist y ∈ S, z ∈ S such that y ∗ x = e and x ∗ z = e, then y = z and y is a ∗-inverse for x. In particular, any ∗-inverse for x is unique.

Notation 6.12. Usually it is more convenient to call a ∗-inverse just an “inverse,” and if an element x has a unique inverse, it can be denoted x^{−1}. There may be some other abbreviations for certain operations; customarily a +-inverse of x is denoted −x.

Exercise 6.13. Given (S, ∗), and x, y ∈ S, if ∗ is associative, and x and y both have ∗-inverses, then x ∗ y has a unique ∗-inverse, y^{−1} ∗ x^{−1}.

Exercise 6.14. Given (S, ∗), and x ∈ S, if ∗ is associative, and there exists a ∗-inverse for x, and x ∗ x = x, then x = e.

Example 6.15. Given a set S, let F denote the set of functions {α : S ⇝ S}. Composition of functions is an example of a binary operation on F, denoted ∘, so that the function α ∘ β is defined by the formula depending on x ∈ S:

(6.1)  (α ∘ β)(x) = α(β(x)).

The operation ∘ is associative and has an identity element denoted Id_{S} ∈ F, which is the function with graph G = {(x, x) : x ∈ S}, so that Id_{S}(x) = x for all x ∈ S.

Notation 6.16. The same symbol ∘ is used for composites of functions between other sets, although this is no longer an example of a binary operation as in Definition 6.3. For any sets S, T, U, and any functions β : S ⇝ T and α : T ⇝ U, there is a composite function α ∘ β : S ⇝ U, defined as in (6.1). An associative property holds: (α ∘ β) ∘ γ = α ∘ (β ∘ γ).

Exercise 6.17. Given S ≠ Ø, and a function α : S ⇝ T, the following are equivalent.

(1) For all s_{1}, s_{2} ∈ S, if s_{1} ≠ s_{2}, then α(s_{1}) ≠ α(s_{2}) (α has the \underline{one-to-one} property).

(2) For any set C and any functions γ : C ⇝ S, δ : C ⇝ S, if α∘γ : C ⇝ T and α∘δ : C ⇝ T are the same function, then γ = δ (α has the \underline{left cancellable} property).

(3) There is a function β : T ⇝ S so that β ∘α : S ⇝ S is equal to the identity function Id_{S} : S ⇝ S (α has a \underline{left inverse}).

Exercise 6.18. Given a function α : S ⇝ T, the following are equivalent.

(1) For all t ∈ T, there is some s ∈ S so that α(s) = t (α has the \underline{onto} property).

(2) For any set C and any functions γ : T ⇝ C, δ : T ⇝ C, if γ∘α : S ⇝ C and δ∘α : S ⇝ C are the same function, then γ = δ (α has the \underline{right cancellable} property).

(3) There is a function β : T ⇝ S so that α ∘ β : T ⇝ T is equal to the identity function Id_{T} : T ⇝ T (α has a \underline{right inverse}).

Hint. The (1) ⟹ (3) step requires the Axiom of Choice.

Exercise 6.19. If β : T ⇝ S is a left inverse of α and γ : T ⇝ S is a right inverse of α then β = γ.

Hint. This is analogous to Exercise 6.11.

Exercise 6.20. Given S ≠ Ø and a function α : S ⇝ T, the following are equivalent.

(1) α is both one-to-one and onto.

(2) α has a left inverse γ : T ⇝ S and a right inverse β : T ⇝ S.

(3) There exists a function δ : T ⇝ S so that α ∘ δ = Id_{T} and δ ∘ α = Id_{S}.

Hint. The equivalence (1) ⟺ (2) uses Exercise 6.17 and Exercise 6.18, although with the one-to-one assumption in (1), the Axiom of Choice is no longer required to construct the right inverse in (2). (3) ⟹ (2) is trivial, and the converse uses Exercise 6.19 to get δ = β = γ.

Definition 6.21. A function α : S ⇝ T is \underline{invertible} means that α satisfies any of the equivalent properties (1), (2), or (3) from Exercise 6.20.

Notation 6.22. A function (such as δ = β = γ as in Exercise 6.19), that is both a left inverse and a right inverse of the function α : S ⇝ T is an \underline{inverse} of α. If an inverse of α exists, then it is unique by Exercise 6.19, it can be denoted α^{−1} : T ⇝ U, and α^{−1} is also invertible, with inverse α.

6.2. Appendix: Quotient spaces

Definition 6.23. Given any vector space (V, +_{V} , ·_{V}) and a subspace W, for any element v ∈ V the following subset of V is called a \underline{coset} of W:

v + W = {v +_{V} w : w ∈ W }.

Definition 6.24. Given a subspace W of V as in Definition 6.23, the set of cosets of W is a vector space with the following operations:

(v_{1} + W) + (v_{2} + W) = (v_{1} +_{V} v_{2}) + W,
ρ · (v + W) = (ρ ·_{V} v) + W,

and zero element 0_{V} + W = W. This vector space is the \underline{quotient space}, denoted V/W.

Exercise 6.25. The function π : V → V/W : v ↦ v + W is linear and right cancellable. If S is a subset of V such that V = span(S), then V/W = span(π(S)).

Exercise 6.26. Given a subspace W of V as in Definition 6.23, and any set S, if B : V ⇝ S is constant on each coset v + W then there exists a unique function b : V/W ⇝ S such that b ∘ π = B.

Hint. For v + W ∈ V/W, choose any element x = v + w ∈ v + W. Define b(v + W) = B(x); this does not depend on the choice of x by hypothesis, and v ∈ v + W (because 0_{V} ∈ W) so b(v + W) = B(x) = B(v). Then for any v ∈ V, (b ∘ π)(v) = b(π(v)) = b(v + W) = B(x) = B(v) as claimed. For uniqueness, use the right cancellable property of π.

Exercise 6.27. Given a subspace W of V as in Definition 6.23, and another vector space U, if B : V → U is linear and satisfies B(w) = 0_{U} for all w ∈ W, then there exists a unique function b : V/W → U such that b ∘ π = B, and b is linear.

Hint. For any v ∈ V, if x ∈ v + W then x = v + w for some w ∈ W and B(x) = B(v + w) = B(v) + B(w) = B(v) + 0_{U} = B(v). So, B is constant on the coset v + W, and the previous Exercise applies to show there is a unique b with b ∘ π = B. The linearity of b easily follows from the linearity of B.

6.3. Appendix: Construction of the tensor product

As mentioned in Section 1.2, we elaborate on the existence of a tensor product of two vector spaces. The notation and methods in this Appendix are specific to this construction and not widely used in the Chapters. We start with a set of functions F(S, W) only because it comes with a convenient vector space structure.

Example 6.28. For any set S ≠Ø and any vector space W, the set of functions

F(S, W) = {f : S ⇝ W }

is a vector space, with the usual operations of pointwise addition of functions and scalar multiplication of functions, and zero element given by the constant function f(x) ≡ 0_{W}.

Notation 6.29. For a set S ≠Ø and any field 𝕂, for each x ∈ S there is an element δ(x) ∈ F(S, 𝕂) defined by:

(6.2)  δ(x) : y ↦ 1 for y = x,
δ(x) : y ↦ 0 for y ≠ x.

The span of the set of such functions is denoted

F_{0}(S, 𝕂) = span ({δ(x) : x ∈ S}) ,

so F_{0}(S, 𝕂) is a subspace of F(S, 𝕂). The notation (6.2) defines a function

δ : S ⇝ F_{0}(S, 𝕂) : x ↦ δ(x).

Exercise 6.30. For S as above and any function f : S ⇝ 𝕂, the following are equivalent.

(1) f ∈ F_{0}(S, 𝕂).

(2) The function f is uniquely expressible as a finite sum of functions with

coefficients α_{ν} ∈ 𝕂 and x_{ν} ∈ S: f = ∑_{ν=1}^{N} α_{ν} · δ(x_{ν}).

(3) f(x) = 0 for all but finitely many x ∈ S.

Exercise 6.31. Given S, 𝕂, and δ as in Notation 6.29, and any vector space W, If g : S ⇝ W is any function, then there exists a unique linear map ḡ : F_{0}(S, 𝕂) → W such that ḡ ∘ δ = g : S ⇝ W.

Hint. As in Exercise 6.30, the general element f of F_{0}(S, 𝕂) has a unique

expression of the form f = ∑_{ν=1}^{N} α_{ν} · δ(x_{ν}); define ḡ on such an expression by using

the same coefficients α_{ν} and elements x_{ν}:

ḡ(f) = ∑_{ν=1}^{N} α_{ν} · g(x_{ν}).

Then for each x ∈ S, the composite function satisfies the claim:

(ḡ ∘ δ)(x) = ḡ(1 · δ(x)) = 1 · g(x) = g(x).

The uniqueness and 𝕂-linearity of ḡ are easily checked.

Definition 6.32. Given vector spaces U and V, define the following subsets of F_{0}(U × V, 𝕂):

R_{1} = {δ((u_{1} + u_{2}, v)) − δ((u_{1}, v)) − δ((u_{2}, v)) : u_{1}, u_{2} ∈ U, v ∈ V }
R_{2} = {δ((u, v_{1} + v_{2})) − δ((u, v_{1})) − δ((u, v_{2})) : u ∈ U, v_{1}, v_{2} ∈ V }
R_{3} = {δ((ρ · u, v)) − ρ · δ((u, v)) : ρ ∈ 𝕂, u ∈ U, v ∈ V }
R_{4} = {δ((u, ρ · v)) − ρ · δ((u, v)) : ρ ∈ 𝕂, u ∈ U, v ∈ V }
R = span (R_{1} ∪ R_{2} ∪ R_{3} ∪ R_{4}) .

The \underline{tensor product space} of U and V is defined to be the quotient space:

U ⊗ V = F_{0}(U × V, 𝕂)/R.

Let π : F_{0}(U × V, 𝕂) → U ⊗ V denote the quotient map as in Exercise 6.25.

Definition 6.33. Define a function τ : U × V ⇝ U ⊗ V by:

τ = π ∘ δ : (u, v) ↦ π(δ((u, v))) = δ((u, v)) + R.

The output τ ((u, v)) is abbreviated u ⊗ v ∈ U ⊗ V.

Exercise 6.34. τ : U × V ⇝ U ⊗ V is a bilinear function.

Hint. Definition 1.23 is easily checked.

Theorem 6.35. For any bilinear function A : U × V ⇝ W, there exists a unique linear map a : U ⊗ V → W such that A = a ∘ τ.

Proof. By Exercise 6.31, there exists a unique linear map Ā : F_{0}(U ×V, 𝕂) → W such that Ā ∘ δ = A. The linear map Ā has value 0_{W} on every element of the subspace R; it is enough to check that Ā(r) = 0_{W} for r in each of the four subsets R_{1}, …, R_{4} from Definition 6.32, for example, for r ∈ R_{1},

Ā(r) = Ā(δ((u_{1} + u_{2}, v)) − δ((u_{1}, v)) − δ((u_{2}, v)))
= A((u_{1} + u_{2}, v)) − A((u_{1}, v)) − A((u_{2}, v))
= 0_{W} .

The other Ā(r) values follow similarly from the bilinear property of A. Exercise 6.27 applies to Ā, to give a unique linear map

a : F_{0}(U × V, 𝕂)/R → W : f + R ↦ Ā(f)

such that a ∘ π = Ā. The conclusion is that

a ∘ τ = a ∘ (π ∘ δ) = (a ∘ π) ∘ δ = Ā ∘ δ = A ⟹ a(u ⊗ v) = A(u, v).

For the uniqueness, suppose there is some a′ with a ∘ π ∘ δ = A = a′ ∘ π ∘ δ. The set {δ((u, v)) : u ∈ U, v ∈ V } spans F_{0}(U × V, 𝕂) as in Notation 6.29, and the image under π of this set, {u ⊗ v : u ∈ U, v ∈ V }, spans U ⊗ V = F_{0}(U × V, 𝕂)/R by Exercise 6.25. So a and a′ agree on a spanning set of U ⊗ V and must be equal.

6.4. APPENDIX: COMMENTS ON [C_{2}] 239

6.4. Appendix: Comments on [C_{2}]

6.4.1. Errata. The following typo appears in the published paper, [C_{2}].

On page 535, line 3, the symbol should be x_{q′} ↦ instead of x_{q′} =.

6.4.2. Updates. The topic of defining a “trace without duals” (as in [C_{2}] §3) is briefly considered by [S] §1.7.

Some details omitted from [C_{2}] are presented here in Chapter 2 (see Remark 2.110) and Section 5.3 (see Remark 5.121).

Bibliography

[AF] F. Anderson and K. Fuller, Rings and Categories of Modules, GTM 13, Springer-Verlag, New York, 1974. MR0417223 (54 #5281)

[AS^{2}] J. Avron, R. Seiler, and B. Simon, The index of a pair of projections, J. of

Functional Analysis (1) 120 (1994), 220–237. MR1262254 (95b:47012)

[Bhatia] R. Bhatia, Matrix Analysis, GTM 169, Springer-Verlag, New York, 1997. MR1477662 (98i:15003)

[B] N. Bourbaki, Algebra I, Chapters 1–3, Hermann, Paris, 1974. MR0354207 (50 #6689)

[C_{1}] A. Coffman, Real linear maps preserving some complex subspaces, Beiträge

zur Algebra und Geometrie (1) 56 (2015), 159–173. MR3305441

[C_{2}] A. Coffman, A non-unital generalized trace and linear complex structures,

Operators and Matrices (2) 15 (2021), 525–569. MR4280043

[CHL] A. Conner, A. Harper, and J. Landsberg, New lower bounds for matrix multiplication and det_{3}, Forum Math. Pi 11 (2023), Paper No. e17, 30 pp. MR4595287

[Cox] D. Cox, Galois Theory, second ed., Wiley, Hoboken, 2012. MR2919975

[DFN] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry — Methods and Applications Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, GTM 93, Springer-Verlag, New York, 1984. MR0736837 (85a:53003)

[EPW] D. Eisenbud, S. Popescu, and C. Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J. (3) 107 (2001), 427–467. MR1828297 (2002d:14075)

[EHM] P. Ellia, A. Hirschowitz, and L. Manivel, Problème de Brill-Noether pour les fibrés de Steiner. Applications aux courbes gauches, Ann. Sci. École Norm. Sup. (4) (no. 6) 32 (1999), 835–857. MR1717579 (2001g:14068)

[FH] W. Fulton and J. Harris, Representation Theory, GTM 129, Springer-Verlag, New York, 1991. MR1153249 (93a:20069)

[GM] T. Garrity and R. Mizner, Invariants of vector-valued bilinear and sesquilinear forms, Linear Algebra Appl. 218 (1995), 225–237. MR1324060 (96d:15044)

[Geroch] R. Geroch, Mathematical Physics, University of Chicago, 1985. MR0824481 (87d:00002)

[AFMC] R. Gillman (ed.), A Friendly Mathematics Competition. 35 Years of Teamwork in Indiana, MAA Problem Books Series 8. MAA Press: An Imprint of the American Mathematical Society, 2003.

[Graham] A. Graham, Kronecker Products and Matrix Calculus: with Applications, EHSMA, Halsted Press, New York, 1981. MR0640865 (83g:15001)

[G_{1}] W. Greub, Linear Algebra, 3rd ed., GMW 97, Springer-Verlag, New York,

1967. MR0224622 (37 #221)

[G_{2}] W. Greub, Multilinear Algebra, GMW 136, Springer-Verlag, New York, 1967.

MR0224623 (37 #222)

[HS] H. Henderson and S. Searle, The vec-permutation matrix, the vec operator and Kronecker products: a review, Linear and Multilinear Algebra (4) 9 (1980/81), 271–288. MR0611262 (82h:15019)

[HK] K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice-Hall, New Jersey, 1971. MR0276251 (43 #1998)

[Hörmander] L. Hörmander, An Introduction to Complex Analysis in Several Variables,

USHM Van Nostrand, Princeton, 1966. MR0203075 (34 #2933)

[HJ] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge, 1991. MR1091716 (92e:15003)

[Humphreys] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Fifth Printing, GTM 9, Springer-Verlag, New York, 1972. MR0499562 (81b:17007)

[J] N. Jacobson, Basic Algebra I, Freeman & Co., San Francisco, 1974. MR0356989 (50 #9457)

[JSV] A. Joyal, R. Street, and D. Verity, Traced monoidal categories, Math. Proc. Camb. Phil. Soc. (3) 119 (1996), 447–468. MR1357057 (96m:18014)

[K] C. Kassel, Quantum Groups, GTM 155, Springer-Verlag, New York, 1995. MR1321145 (96e:17041)

[LP] A. Lascoux and P. Pragacz, Schur Q-functions and degeneracy locus formulas for morphisms with symmetries, in Recent Progress in Intersection Theory (Bologna, 1997), Trends Math., Birkhäuser, Boston, 2000, 239–263. MR1849297 (2002g:14004)

[L] D. W. Lewis, Trace forms, Kronecker sums, and the shuffle matrix, Linear and Multilinear Algebra 40 (1996), 221–227. MR1382078 (97h:15038)

[MB] S. Mac Lane and G. Birkhoff, Algebra, 3rd ed., Chelsea, New York, 1993. MR0941522 (89i:00009)

[Magnus] J. Magnus, Linear Structures, Griffin’s Statistical Monographs and Courses 42, Oxford University Press, New York, 1988. MR0947343 (89j:15003)

[Maltsiniotis] G. Maltsiniotis, Traces dans les catégories monoïdales, dualité et catégories monoïdales fibrées, Cahiers Topologie Géom. Différentielle Catég. (3) 36 (1995), 195–288. MR1352535 (97d:18007)

[Neudecker] H. Neudecker, Some theorems on matrix differentiation with special reference to Kronecker matrix products, Journal of the American Statistical Association 64 (1969), 953–963.

[Nissen] D. H. Nissen, A note on the variance of a matrix, Econometrica (3 – 4) 36 (1968), 603–604.

[O] I. Ojeda, Kronecker square roots and the block vec matrix, The American Mathematical Monthly (1) 122 (2015), 60–64. MR3324956

[P] T. Pate, Tensor products, symmetric products, and permanents of positive semidefinite Hermitian matrices, Linear and Multilinear Algebra (1 – 4) 31 (1992), 27–36. MR1199038 (94a:15048)

[PS] K. Ponto and M. Shulman, Shadows and traces in bicategories, J. Homotopy Relat. Struct. (2) 8 (2013), 151–200. MR3095324

[R] W. Roth, On direct product matrices, Bull. Amer. Math. Soc. (6) 40 (1934), 461–468. MR1562881

[Stolz-Teichner] S. Stolz and P. Teichner, Traces in monoidal categories, Trans. Amer. Math.

Soc. (8) 364 (2012), 4425–4464. MR2912459

[S] R. Street, Monoidal categories in, and linking, geometry and algebra, Bull. Belg. Math. Soc. Simon Stevin (5) 19 (2012), 769–821. MR3009017

[Sukhov-Tumanov] A. Sukhov and A. Tumanov, Regularization of almost complex structures and gluing holomorphic discs to tori, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (no. 2) 10 (2011), 389–411. MR2856153 (2012m:32031)

[VZ] C. Vafa, E. Zaslow, et al, eds., Mirror Symmetry, Clay Math. Monog. 1, AMS, Providence, 2003. MR2003030 (2004g:14042)