Category of Super Vector Spaces Becomes a Tensor Category

The theory of manifolds and algebraic geometry are ultimately based on linear algebra. Similarly the theory of supermanifolds needs super linear algebra, which is linear algebra in which vector spaces are replaced by vector spaces with a Z/2Z-grading, namely, super vector spaces.

A super vector space is a Z/2Z-graded vector space

V = V₀ ⊕ V₁

where the elements of V₀ are called even and that of V₁ odd.

The parity of v ∈ V , denoted by p(v) or |v|, is defined only on non-zero homogeneous elements, that is elements of either V₀ or V₁:

p(v) = |v| = 0 if v ∈ V₀

= 1 if v ∈ V₁

The superdimension of a super vector space V is the pair (p, q) where dim(V₀) = p and dim(V₁) = q as ordinary vector spaces. We simply write dim(V) = p|q.

If dim(V) = p|q, then we can find a basis {e₁,…., e_p} of V₀ and a basis {ε₁,….., ε_q} of V₁ so that V is canonically isomorphic to the free k-module generated by {e₁,…., e_p, ε₁,….., ε_q}. We denote this k-module by k^p|q and we will call {e₁,…., e_p, ε₁,….., ε_q} the canonical basis of k^p|q. The (e_i) form a basis of k^p = k₀^p|q and the (ε_j) form a basis for k^q = k₁^p|q.

A morphism from a super vector space V to a super vector space W is a linear map from V to W preserving the Z/2Z-grading. Let Hom(V, W) denote the vector space of morphisms V → W. Thus we have formed the category of super vector spaces that we denote by (smod). It is important to note that the category of super vector spaces also admits an “inner Hom”, which we denote by Hom(V, W); for super vector spaces V, W, Hom(V, W) consists of all linear maps from V to W ; it is made into a super vector space itself by:

Hom(V, W)₀ = {T : V → W|T preserves parity}  (= Hom(V, W))

Hom(V, W)₁ = {T : V → W|T reverses parity}

If V = k^m|n, W = k^p|q we have in the canonical basis (e_i, ε_j):

Hom(V, W)₀ = (^{A 0} _{0 D}) and Hom(V, W)₁ = (^{0 B} _{C 0})

where A, B, C , D are respectively (p x m), (p x n), (q x m), (q x n) – matrices with entries in k.

In the category of super vector spaces we have the parity reversing functor ∏(V → ∏V) defined by

(∏V)₀ = V₁, (∏V)₁ = V₀

The category of super vector spaces admits tensor products: for super vector spaces V, W, V ⊗ W is given the Z/2Z-grading as

(V ⊗ W)₀ = (V₀ ⊗ W₀) ⊕ (V₁ ⊗ W₁),

(V ⊗ W)₁ = (V₀ ⊗ W₁) ⊕ (V₁ ⊗ W₀)

The assignment V, W ↦ V ⊗ W is additive and exact in each variable as in the ordinary vector space category. The object k functions as a unit element with respect to tensor multiplication ⊗ and tensor multiplication is associative, i.e., the two products U ⊗ (V ⊗ W) and (U ⊗ V) ⊗ W are naturally isomorphic. Moreover, V ⊗ W ≅ W ⊗ V by the commutative map,

c_V,W : V ⊗ W → W ⊗ V

where

v ⊗ w ↦ (-1)^|v||w|w ⊗ v

If we are working with the category of vector spaces, the commutativity isomorphism takes v ⊗ w to w ⊗ v. In super linear algebra we have to add the sign factor in front. This is a special case of the general principle called the “sign rule”. The principle says that in making definitions and proving theorems, the transition from the usual theory to the super theory is often made by just simply following this principle, which introduces a sign factor whenever one reverses the order of two odd elements. The functoriality underlying the constructions makes sure that the definitions are all consistent.

The commutativity isomorphism satisfies the so-called hexagon diagram:

where, if we had not suppressed the arrows of the associativity morphisms, the diagram would have the shape of a hexagon.

The definition of the commutativity isomorphism, also informally referred to as the sign rule, has the following very important consequence. If V1, …, Vn are the super vector spaces and σ and τ are two permutations of n-elements, no matter how we compose associativity and commutativity morphisms, we always obtain the same isomorphism from V_σ(1) ⊗ … ⊗ V_σ(n) to V_τ(1) ⊗ … ⊗ V_τ(n) namely:

V_σ(1) ⊗ … ⊗ V_σ(n) → V_τ(1) ⊗ … ⊗ V_τ(n)

_vσ(1) ⊗ … ⊗ v_σ(n) ↦ (-1)^N v_τ(1) ⊗ … ⊗ v_τ(n)

where N is the number of pair of indices i, j such that v_i and v_j are odd and σ^-1(i) < σ^-1(j) with τ^-1(i) > τ^-1(j).

The dual V* of V is defined as

V* := Hom (V, k)

If V is even, V = V₀, V* is the ordinary dual of V consisting of all even morphisms V → k. If V is odd, V = V₁, then V* is also an odd vector space and consists of all odd morphisms V¹ → k. This is because any morphism from V₁ to k = k^1|0 is necessarily odd and sends odd vectors into even ones. The category of super vector spaces thus becomes what is known as a tensor category with inner Hom and dual.

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