These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.
The parity of a nonzero homogeneous element, denoted by
Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity.
The standard super coordinate space, denoted
coordinate basis vectors and the odd space is spanned by the last
Homogeneous subspaces are super vector spaces in their own right (with the obvious grading).
to be the super vector space with the even and odd subspaces interchanged.
That is, A homomorphism, a morphism in the category of super vector spaces, from one super vector space to another is a grade-preserving linear transformation.
between super vector spaces is grade preserving if That is, it maps the even elements of
An isomorphism of super vector spaces is a bijective homomorphism.
, so that The usual algebraic constructions for ordinary vector spaces have their counterpart in the super vector space setting.
can be regarded as a super vector space by taking the even functionals to be those that vanish on
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 One can also construct tensor products of super vector spaces.
The underlying space is as in the ungraded case with the grading given by where the indices are in
Specifically, one has 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.
by considering the (graded) tensor product The category of super vector spaces, denoted by
, is the category whose objects are super vector spaces (over a fixed field
) 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.
is a monoidal category with the super tensor product as the monoidal product and the purely even super vector space
The involutive braiding operator given by on homogeneous elements, turns
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.
is also a closed monoidal category with the internal Hom object,
, given by the super vector space of all linear maps from
with a multiplication map that is a super vector space homomorphism.
This is equivalent to demanding[5] Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over