BRST operator for quantum Lie algebras and differential calculus on quantum groups

TL;DR

This paper constructs BRST operators for quantum Lie algebras and develops a differential calculus on quantum groups, extending classical geometric concepts to the quantum setting with explicit operators and a Hodge decomposition.

Contribution

It introduces a novel BRST operator framework for quantum Lie algebras and formulates a differential calculus on quantum groups, including explicit constructions and a Hodge decomposition.

Findings

Explicit BRST and anti-BRST operators are constructed.

A recurrent relation uniquely defines the BRST operator Q.

Hodge decomposition theorem is formulated for U_q(gl(N)).

Abstract

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)).