This article is cited in 16 papers

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

A. P. Isaev^a, O. V. Ogievetskii<sup>bc

a</sup> Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics
^b P. N. Lebedev Physical Institute, Russian Academy of Sciences
^c CNRS – Center of Theoretical Physics

Abstract: 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 $U_{\mathrm q}(gl(N))$.

DOI: 10.4213/tmf537

 English version: Theoretical and Mathematical Physics, 2001, 129:2, 1558–1572

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