Notes on the differential calculi on quantum linear groups

This talk is devoted to the problem of construction of the differential calculi on quantum linear groups. Basing on the natural algebraic postulates we examine the possible commutation relations for the $GL_q(N)$- and $SL_q(N)$-invariant differential forms and vector fields. It turns out that there exist several families of the admissible commutation rules for $GL_q(N)$, but, in contrast, the commutation prescription for $SL_q(N)$ is unique.