Supermatrix - Definitions and Notation

Definitions and Notation

Let R be a fixed superalgebra (assumed to be unital and associative). Often one requires R be supercommutative as well (for essentially the same reasons as in the ungraded case).

Let p, q, r, and s be nonnegative integers. A supermatrix of dimension (r|s)×(p|q) is a matrix with entries in R that is partitioned into a 2×2 block structure

with r+s total rows and p+q total columns (so that the submatrix X₀₀ has dimensions r×p and X₁₁ has dimensions s×q). An ordinary (ungraded) matrix can be thought of as a supermatrix for which q and s are both zero.

A square supermatrix is one for which (r|s) = (p|q). This means that not only is the unpartitioned matrix X square, but the diagonal blocks X₀₀ and X₁₁ are as well.

An even supermatrix is one for which diagonal blocks (X₀₀ and X₁₁) consist solely of even elements of R (i.e. homogeneous elements of parity 0) and the off-diagonal blocks (X₀₁ and X₁₀) consist solely of odd elements of R.

An odd supermatrix is one for the reverse holds: the diagonal blocks are odd and the off-diagonal blocks are even.

If the scalars R are purely even there are no nonzero odd elements, so the even supermatices are the block diagonal ones and the odd supermatrices are the off-diagonal ones.

A supermatrix is homogeneous if it is either even or odd. The parity, |X|, of a nonzero homogeneous supermatrix X is 0 or 1 according to whether it is even or odd. Every supermatrix can be written uniquely as the sum of an even supermatrix and an odd one.