Quantization of quasi-Lie bialgebras

Enriquez, Benjamin; Halbout, Gilles

doi:10.1090/S0894-0347-10-00654-5

Quantization of quasi-Lie bialgebras
HTML articles powered by AMS MathViewer

by Benjamin Enriquez and Gilles Halbout;

J. Amer. Math. Soc. 23 (2010), 611-653

DOI: https://doi.org/10.1090/S0894-0347-10-00654-5

Published electronically: January 15, 2010

PDF | Request permission

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.

References

A. Alekseev, C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, preprint arXiv:0802:4300.
N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1349, Hermann, Paris, 1972. MR 573068
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
V. G. Drinfel′d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR 1047964
V. G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 829–860. MR 1080203
V. G. Drinfel′d, On some unsolved problems in quantum group theory, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 1–8. MR 1183474, DOI 10.1007/BFb0101175
B. Enriquez, On some universal algebras associated to the category of Lie bialgebras, Adv. Math. 164 (2001), no. 1, 1–23. MR 1870510, DOI 10.1006/aima.2001.2007
B. Enriquez, G. Halbout, Quantization of coboundary Lie bialgebras, preprint arXiv:math/0603740, to appear in Annals of Math.
Pavel Etingof and David Kazhdan, Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), no. 1, 1–41. MR 1403351, DOI 10.1007/BF01587938
Pavel Etingof and David Kazhdan, Quantization of Lie bialgebras. II, III, Selecta Math. (N.S.) 4 (1998), no. 2, 213–231, 233–269. MR 1669953, DOI 10.1007/s000290050030
Fabio Gavarini, The quantum duality principle, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 3, 809–834 (English, with English and French summaries). MR 1907388, DOI 10.5802/aif.1902
Murray Gerstenhaber and Samuel D. Schack, Bialgebra cohomology, deformations, and quantum groups, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 1, 478–481. MR 1031952, DOI 10.1073/pnas.87.1.478
Bertram Kostant and Shlomo Sternberg, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176 (1987), no. 1, 49–113. MR 893479, DOI 10.1016/0003-4916(87)90178-3
Pierre B. A. Lecomte and Claude Roger, Modules et cohomologies des bigèbres de Lie, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 405–410 (French, with English summary). MR 1046522
Saunders Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40–106. MR 171826, DOI 10.1090/S0002-9904-1965-11234-4
L. Positselski, letter to M. Finkelberg and R. Bezrukavnikov (in Russian), 1995.
Steven Shnider and Shlomo Sternberg, Quantum groups, Graduate Texts in Mathematical Physics, II, International Press, Cambridge, MA, 1993. From coalgebras to Drinfel′d algebras; A guided tour. MR 1287162