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Quantization of quasi-Lie bialgebras

Authors: Benjamin Enriquez and Gilles Halbout
Journal: J. Amer. Math. Soc. 23 (2010), 611-653
MSC (2010): Primary 18D10, 17B37
Published electronically: January 15, 2010
MathSciNet review: 2629982
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Abstract: We construct quantization functors of quasi-Lie bialgebras. We establish a bijection between this set of quantization functors, modulo equivalence and twist equivalence, and the set of quantization functors of Lie bialgebras, modulo equivalence. This is based on the acyclicity of the kernels of natural morphisms between the universal versions of Lie algebra cohomology complexes for quasi-Lie and Lie bialgebras. The proof of this acyclicity consists of several steps, ending up in the acyclicity of a complex related to free Lie algebras, namely, the universal version of the Lie algebra cohomology complex of a Lie algebra in its enveloping algebra, viewed as the left regular module. Using the same arguments, we also prove the compatibility of quantization functors of quasi-Lie bialgebras with twists, which allows us to recover our earlier results on compatibility of quantization functors with twists in the case of Lie bialgebras.

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Additional Information

Benjamin Enriquez
Affiliation: Institut de Recherche Matématique Avancée (CNRS) et Université de Strasbourg, 7, rue René Descartes, F-67084 Strasbourg, France

Gilles Halbout
Affiliation: Institut de Mathématiques, Université Montpellier 2, Place E. Bataillon, F-34095 Montpellier, France

Received by editor(s): April 10, 2008
Published electronically: January 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.