In mathematics, a super vector space is a \mathbb Z_2-graded vector space, that is, a vector space over a field with a given decomposition. 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

-graded vector space with decomposition

are said to be homogeneous. The parity,

are called even 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.

is finite-dimensional is said to have dimension, is the ordinary coordinate space.

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).

, one can define the parity reversed space 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 between super vector spaces is grade preserving if

. An isomorphism of super vector spaces is a bijective.

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 such that

Declaring the grade-preserving transformations to be even and the grade-reversing ones to be odd and called internal, the structure of a super vector space. In particular,

, 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 can be regarded as a super vector space by taking the even functionals. 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 comes into play. The underlying space is as in the ungraded case with the grading given by

. Specifically, one has

Supermodules

Just as one may generalize vector spaces over a field to moduless 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 let

denote the Grassmann algebra generated by considering the (graded) tensor product

The category of super vector spaces

The category of super vector spaces, is the category whose objects) 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, supergroupss, etc. that is completely analogous to their ungraded counterparts.

is a monoidal category as the unit object. The involutive braiding operator

given by

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.

is also a closed monoidal category with the internal Hom object, given by the super vector space of all is the even subspace therein:

is left adjoint, given a natural bijection

Superalgebra

A superalgebra 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 is a monoid.

Notes

References