BRST Operator for Quantum Lie Algebras: Relation to Bar Complex

TL;DR

This paper explores the relationship between the BRST operator and the bar complex in quantum Lie algebras, establishing an isomorphism between standard and antisymmetric chain complexes and extending to graded cases.

Contribution

It constructs a chain map linking the standard complex of quantum Lie algebras to the antisymmetric subcomplex, generalizing the complex to graded quantum Lie algebras.

Findings

Established an isomorphism between the standard complex and the antisymmetric chain subcomplex.

Derived nontrivial identities in the braid group algebra essential for the construction.

Extended the complex framework to graded quantum Lie algebras.

Abstract

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) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given and solved. 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 discuss also a generalization of the standard complex to the case when a q-Lie algebra is equipped with a grading operator.