Gauss Decomposition for Quantum Groups and Supergroups

Abstract

The Gauss decompositions of the quantum groups, related to classical Lie groups and supergroups are considered by the elementary algebraic and $R$-matrix methods. The commutation relations between new basis generators (which are introduced by the decomposition) are described in some details. It is shown that the reduction of the independent generator number in the new basis to the dimension of related classical (super) group is possible. The classical expression for (super) determinant through the Gauss decomposition generators is not changed in the deformed case. The symplectic quantum group $Sp_q(2)$ and supergroups $GL_q(1|1), GL_q(2|1)$ are considered as examples.