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

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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 q(gl(N)).

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Moscow Oblast, Russia

    A. P. Isaev

  2. Centre de Physique Théorique, Luminy, Marseille, France

    O. V. Ogievetsky

  3. Tamm Department of Theoretical Physics, Lebedev Physical Institute, RAS, Moscow, Russia

    O. V. Ogievetsky

Authors
  1. A. P. Isaev
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  2. O. V. Ogievetsky
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Isaev, A.P., Ogievetsky, O.V. BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups. Theoretical and Mathematical Physics 129, 1558–1572 (2001). https://doi.org/10.1023/A:1012839308392

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