In mathematics, a super vector space is a
Contents
Definitions
A super vector space is a
Vectors that are elements of either
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
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
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
That is, it maps the even elements of
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
Declaring the grade-preserving transformations to be even and the grade-reversing ones to be odd gives the space of all linear transformations from
A grade-reversing transformation from
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
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
where the indices are in
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
denote the Grassmann algebra generated by
The category of super vector spaces
The category of super vector spaces, denoted by
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
given by
on homogeneous elements, turns
The fact that
Superalgebra
A superalgebra over
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