Super vector space

In mathematics, a super vector space is a formula_1-graded vector space, that is, a vector space over a field formula_2 with a given decomposition of subspaces of grade formula_3 and grade formula_4. 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.
A super vector space is a formula_1-graded vector space with decomposition
Vectors that are elements of either formula_7 or formula_8 are said to be "homogeneous". The "parity" of a nonzero homogeneous element, denoted by formula_9, is formula_3 or formula_4 according to whether it is in formula_7 or formula_8,
Vectors of parity 0 are called "even" and those of parity 1 are called "odd". In theoretical physics, the even elements are sometimes called "Bose elements" or "bosonic", and the odd elements "Fermi elements" or fermionic. Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity.
If formula_15 is finite-dimensional and the dimensions of formula_7 and formula_8 are formula_18 and formula_19 respectively, then formula_15 is said to have "dimension" formula_21. The standard super coordinate space, denoted formula_22, is the ordinary coordinate space formula_23 where the even subspace is spanned by the first formula_18 coordinate basis vectors and the odd space is spanned by the last formula_19.
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 formula_15, one can define the "parity reversed space" formula_27 to be the super vector space with the even and odd subspaces interchanged. That is,
Linear transformations.
A homomorphism, a morphism in the category of super vector spaces, from one super vector space to another is a grade-preserving linear transformation. A linear transformation formula_29 between super vector spaces is grade preserving if
That is, it maps the even elements of formula_15 to even elements of formula_32 and odd elements of formula_15 to odd elements of formula_32. An isomorphism of super vector spaces is a bijective homomorphism. The set of all homomorphisms formula_35 is denoted formula_36.
Every linear transformation, not necessarily grade-preserving, 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 formula_29 such that 
Declaring the grade-preserving transformations to be "even" and the grade-reversing ones to be "odd" gives the space of all linear transformations from formula_15 to formula_32, denoted formula_41 and called "internal" formula_42, the structure of a super vector space. In particular,
A grade-reversing transformation from formula_15 to formula_32 can be regarded as a homomorphism from formula_15 to the parity reversed space formula_47, so that
Operations on super vector spaces.
The usual algebraic constructions for ordinary vector spaces have their counterpart in the super vector space setting.
Dual space.
The dual space formula_49 of a super vector space formula_15 can be regarded as a super vector space by taking the even functionals to be those that vanish on formula_8 and the odd functionals to be those that vanish on formula_7. Equivalently, one can define formula_49 to be the space of linear maps from formula_15 to formula_55 (the base field formula_2 thought of as a purely even super vector space) with the gradation given in the previous section.
Direct sum.
Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by
Tensor product.
One can also construct tensor products of super vector spaces. Here the additive structure of formula_1 comes into play. The underlying space is as in the ungraded case with the grading given by
where the indices are in formula_1. Specifically, one has
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 formula_2 let
denote the Grassmann algebra generated by formula_66 anticommuting odd elements formula_67. Any super vector formula_15 space over formula_2 can be embedded in a module over formula_70 by considering the (graded) tensor product
The category of super vector spaces.
The category of super vector spaces, denoted by formula_72, is the category whose objects are super vector spaces (over a fixed field formula_2) 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 formula_72 is a monoidal category with the super tensor product as the monoidal product and the purely even super vector space formula_55 as the unit object. The involutive braiding operator
given by
on homogeneous elements, turns formula_72 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.
formula_72 is also a closed monoidal category with the internal Hom object, formula_41, given by the super vector space of "all" linear maps from formula_15 to formula_32. The ordinary formula_42 set formula_36 is the even subspace therein:
The fact that formula_72 is closed means that the functor formula_87 is left adjoint to the functor formula_88, given a natural bijection
Superalgebra.
A superalgebra over formula_2 can be described as a super vector space formula_91 with a multiplication map
that is a super vector space homomorphism. This is equivalent to demanding
Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over formula_2 is a monoid in the category formula_72.