Super Vector Space - Definitions

Definitions

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

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).

For any super vector space V, one can define the parity reversed space ΠV to be the super vector space with the even and odd subspaces interchanged. That is,

\begin{align} (\Pi V)_0 &= V_1 \\ (\Pi V)_1 &= V_0.\end{align}

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