Two parameter deformation of Grassmann matrix group and supergroup

Abstract

The two parameter quantum deformation 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 genel 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 are expressed in terms of an $\hat{R}$-matrix. The properties of $n$th power of a Grassmann supermatrix are given as an Appendix.