Two parameter deformation of Grassmann matrix group and supergroup

Abstract

The two parameter quantum deformations of 2X2 Grassmann matrices, Gr(2), and supermatrices, Gr(1\1), are presented. Gr(2) whose matrix elements are all Grassmannian variables is called the superdual of the general linear group GL(2), and Gr(1\1) whose diagonal matrix elements are Grassmannian variables is called the superdual of the supergroup GL(1\1) whose nondiagonal elements are Grassmannian. Noncentral dual superdeterminant for Grassmann supermatrices belonging to Gr(p,q)(1\1) is constructed. As with the 2X2 quantum matrices, the relations satisfied by the matrix elements of the Grassmann matrices and supermatrices are expressed in terms of an (R) over tilde matrix. The properties of the nth power of a Grassmann supermatrix are given as an Appendix. (C) 1996 American Institute of Physics.