Super vector space

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In mathematics, a super vector space is another name for 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 described the various algebraic aspects of supersymmetry.

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[edit] Definitions

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

|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).

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).

[edit] Generalizations

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 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 Grassman algebra generated by N anticommuting odd elements θi. Any super vector space over K can be regarded as a module over R by taking the (graded) tensor product

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

[edit] The category of super vector spaces

In mathematics, Vect(K, Z2) denotes the category whose objects are "all" of the Z2-graded vector spaces over the given field K. The morphisms of this category are given by the even and odd linear transformations between any two such objects. A linear transformation φ: A → B is said to be even if it maps the even part ([0] mod 2) of A to the even part of B and the odd part ([1] mod 2) of A to the odd part of B, and is said to be odd if it maps the even part of A to the odd part of B and the odd part of A to the even part of B. Note that any linear transformation can then be expressed uniquely as the sum of an even and an odd linear transformation.

Vect(K, Z2) is an example of a Z2-graded category. The even morphisms map to 0 while the odd morphisms map to 1. It is a monoidal category since the tensor product of two Z2-graded vector spaces is another Z2-graded vector space.

It is also a braided monoidal category with the involutive braiding operator

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

given by

\tau_{U,V}(u\otimes v)=(-1)^{|u||v|}v \otimes u

for pure elements. In this sense, it is a symmetric monoidal category.

There is also a parity reversing functor from this category to itself which interchanges the even and odd subspaces.

This category is the foundation for the study of "superobjects" like superalgebras, Lie superalgebras, supergroups, supermanifolds, superspace etc.

We can also grade the vector spaces by more general groups. For example, in the study of anyonic objects, we use the category of Z/NZ-graded vector spaces instead. In this case, the involutive braiding operator is given by

\tau_{U,V}(u\otimes v)= e^{2\pi i |u||v|/N} v\otimes u.

[edit] References

  • Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes in Mathematics 11, American Mathematical Society. ISBN 0-8218-3574-2. 
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