BRST Operator for Quantum Lie Algebras: Relation to the Bar Complex

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit quantum generalizations. In particular, there is a BRST operator $Q$ ($Q^2=0$) that generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers, we gave and solved a recursive relation for the operator $Q$ for quantum Lie algebras. Here, we consider the bar complex for $q$-Lie algebras and its subcomplex of $q$-antisymmetric chains. We establish a chain map (which is an isomorphism) of the standard complex for a $q$-Lie algebra to the subcomplex of the antisymmetric chains. The construction requires a set of nontrivial identities in the group algebra of the braid group. We also discuss a generalization of the standard complex to the case where a $q$-Lie algebra is equipped with a grading operator.