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

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

2001-06-25

arxiv.org/abs/math/0106206v2

Mathematics

Quantum Algebra

20 pages, LaTeX, Lecture given at the Workshop on "Classical and Quantum Integrable Systems", 8 - 11 January, Protvino, Russia

Scientific paper

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)).

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