BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups

For a Hopf algebra $\mathcal A$, we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of $\mathcal A$ and (2) an exterior extension of the dual algebra $\mathcal A^*$. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on $\mathcal A$. The first differential complex is an analogue of the de Rham complex. When $\mathcal A^*$ is a universal enveloping algebra 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 $Q$. We give a recursive relation that uniquely defines the operator $Q$. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra $U_{\mathrm q}(gl(N))$.