Gauss decomposition for quantum groups and supergroups

The Gauss decompositions of quantum groups related to classical Lie groups and supergroups are considered by the elementary algebraic and $R$-matrix methods. The commutation relations between the generators of the new basis introduced by the decomposition are described in detail. It is shown that it is possible to reduce a number of independent generators to the dimension of the related classical group. The symplectic quantum group $Sp_q(2)$ and supergroups $GL_q(1, 1)$ and $GL_q(2, 1)$ are considered as examples. Bibliography: 66 titles.