Super vector space

In mathematics, a super vector space is a Z2-graded vector space, that is, a vector space over a field K with a given decomposition

V=V_0\oplus V_1.

The study of super vector spaces and their generalizations is sometimes called super linear algebra. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.

Definitions

Vectors which are elements of either V0 or V1 are said to be homogeneous. The parity of a nonzero homogeneous element, denoted by |x|, is 0 or 1 according to whether it is in V0 or V1.

|x| = \begin{cases}0 & x\in V_0\\1 & x\in V_1\end{cases}

Vectors of parity 0 are called even and those of parity 1 are called odd. Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity.

If V is finite-dimensional and the dimensions of V0 and V1 are p and q respectively, then V is said to have dimension p|q. The standard super coordinate space, denoted Kp|q, is the ordinary coordinate space Kp+q where the even subspace is spanned by the first p coordinate basis vectors and the odd space is spanned by the last q.

A homogeneous subspace of a super vector space is a linear subspace that is spanned by homogeneous elements. Homogeneous subspaces are super vector spaces in their own right (with the obvious grading).

For any super vector space V, one can define the parity reversed space ΠV to be the super vector space with the even and odd subspaces interchanged. That is,

\begin{align} (\Pi V)_0 &= V_1 \\ (\Pi V)_1 &= V_0.\end{align}

Linear transformations

A homomorphism from one super vector space to another is a grade-preserving linear transformation. A linear transformation f : V → W between super vector spaces is grade preserving if

f(V_i) \sub W_{i}

for i = 0 and 1. That is, it maps the even elements of V to even elements of W and odd elements of V to odd elements of W. An isomorphism of super vector spaces is a bijective homomorphism.

Every linear transformation from one super vector space to another can be written uniquely as the sum of a grade-preserving transformation and a grade-reversing one—that is, a transformation f : V → W such that

f(V_i) \sub W_{1-i}

for i = 0 and 1. Declaring the grade-preserving transformations to be even and the grade-reversing ones to be odd gives the space of all linear transformations from V to W the structure of a super vector space.

Note that a grade-reversing transformation from V to W can be regarded as a homomorphism from V to the parity reversed space ΠW.

Operations on super vector spaces

The dual space V* of a super vector space V can be regarded as a super vector space by taking the even functionals to be those that vanish on V1 and the odd functionals to be those that vanish on V0. Equivalently, one can define V* to be the space of linear maps from V to K1|0 (the base field K thought of as a purely even super vector space) with the gradation given in the previous section.

Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by

(V\oplus W)_0 = V_0\oplus W_0
(V\oplus W)_1 = V_1\oplus W_1.

One can also construct tensor products of super vector spaces. Here the additive structure of Z2 comes into play. The underlying space is as in the ungraded case with the grading given by

(V\otimes W)_i = \bigoplus_{j+k=i}V_j\otimes W_k

where the indices are in Z2. Specifically, one has

(V\otimes W)_0 = (V_0\otimes W_0)\oplus(V_1\otimes W_1),
(V\otimes W)_1 = (V_0\otimes W_1)\oplus(V_1\otimes W_0).

Supermodules

Just as one may generalize vector spaces over a field to modules over a commutative ring, one may generalize super vector spaces over a field to supermodules over a supercommutative algebra (or ring).

A common construction when working with super vector spaces is to enlarge the field of scalars to a supercommutative Grassmann algebra. Given a field K let

R = K[\theta_1, \cdots, \theta_N]

denote the Grassmann algebra generated by N anticommuting odd elements θi. Any super vector space over K can be embedded in a module over R by considering the (graded) tensor product

K[\theta_1, \cdots, \theta_N]\otimes V.

The category of super vector spaces

The category of super vector spaces, denoted by K-SVect, is the category whose objects are super vector spaces (over a fixed field K) and whose morphisms are even linear transformations (i.e. the grade preserving ones).

The categorical approach to super linear algebra is to first formulate definitions and theorems regarding ordinary (ungraded) algebraic objects in the language of category theory and then transfer these directly to the category of super vector spaces. This leads to a treatment of "superobjects" such as superalgebras, Lie superalgebras, supergroups, etc. that is completely analogous to their ungraded counterparts.

The category K-SVect is a monoidal category with the super tensor product as the monoidal product and the purely even super vector space K1|0 as the unit object. The involutive braiding operator

\tau_{V,W}: V\otimes W \rightarrow W\otimes V,

given by

\tau_{V,W}(x\otimes y)=(-1)^{|x||y|}y \otimes x

on pure elements, turns K-SVect into a symmetric monoidal category. This commutativity isomorphism encodes the "rule of signs" that is essential to super linear algebra. It effectively says that a minus sign is picked up whenever two odd elements are interchanged. One need not worry about signs in the categorical setting as long as the above operator is used wherever appropriate.

K-SVect is also a closed monoidal category with the internal Hom object, Hom(V, W), given by the super vector space of all linear maps from V to W. The ordinary Hom set Hom(V, W) is the even subspace therein:

\mathrm{Hom}(V, W) = \mathbf{Hom}(V,W)_0.

The fact that K-SVect is closed means that the functor –⊗V is left adjoint to the functor Hom(V,–), given a natural bijection:

\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U,\mathbf{Hom}(V,W)).

A superalgebra over K can be described as a super vector space A with a multiplication map

\mu : A \otimes A \to A

Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over K is a monoid in the category K-SVect.

References

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