Differential geometry of GL$_{p,q}(1| 1)$

Abstract

We introduce a construction of the differential calculus on the quantum supergroup GL$_{p,q}(1| 1)$. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum superalgebra of GL$_{p,q}(1| 1)$. Although all of the structures we obtain are derived without an R matrix, they neverthless can be expressed using an R matrix.