  For a Hopf algebra A, we define the structures of differential complexes on
two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an
exterior extension of the dual algebra A^*. The Heisenberg double of these two
exterior Hopf algebras defines the differential algebra for the Cartan
differential calculus on A. The first differential complex is an analog of the
de Rham complex. In the situation when A^* is a universal enveloping of a Lie
(super)algebra the second complex coincides with the standard complex. The
differential is realized as an (anti)commutator with a BRST- operator Q. A
recurrent relation which defines uniquely the operator Q is given. The BRST and
anti-BRST operators are constructed explicitly and the Hodge decomposition
theorem is formulated for the case of the quantum Lie algebra U_q(gl(N)).
