Abstract
This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra \(\mathfrak{g}(A)\) corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra \(\mathfrak{g}(A)\) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).
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References
V. Drinfeld, On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, Vol. 1510, Springer-Verlag, New York, 1992, pp. 1–8.
V. Drinfeld, Quantum groups, in: Proceedings of ICM-86 (Berkeley), Vol. 1, 1987, pp. 798–820.
В. Дринфельд, Квазихопфовы алгебры, Алгебра и анализ 1 (1989), no. 6, 114–148. English transl.: V. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), no. 6, 1419–1457.
В. Дринфельд, О квазитреугольных квазихопфовых алгебрах и некоторых группах, близко связанных с \({\text{Gal(}}\overline{{\text{Q}}} {\text{/Q)}}\), Алгебра и анализ 2 (1990), no. 4, 149–181. English transl.: V. Drinfeld, On quasitriangular quasi-Hopf algebras and a certain group closely connected with \({\text{Gal(}}\overline{{\text{Q}}} {\text{/Q)}}\), Leningrad Math. J. 2 (1991), no. 4, 829–860.
B. Enriquez, N. Geer, Compatibility of quantization functors of Lie bialgebras with duality and doubling operations, 2007, math/0707.2337.
P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, I, Selecta Math. 2 (1996), no. 1, 1–41.
P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, II, Selecta Math. 4 (1998), 213–231.
P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, III, Selecta Math. 4 (1998), 233–269.
P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, IV, Selecta Math. 6 (2000), no. 1, 79–104.
P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, V, Selecta Math. 6, no. 1, 105–130.
M. Jimbo, A q-analog of \( U{\left( {\mathfrak{g}\mathfrak{l}{\left( {N + 1} \right)}} \right)} \), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247–252.
V. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge, 1990.
D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, III, J. Amer. Math. Soc. 7 (1994), no. 2, 383–453.
G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, 1994.
A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Quantum Groups, Advanced Series in Mathematical Physics, Vol. 21, World Scientific, River Edge, NJ, 1995.
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To Bert Kostant with admiration
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Etingof, P., Kazhdan, D. Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras. Transformation Groups 13, 527–539 (2008). https://doi.org/10.1007/s00031-008-9029-6
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DOI: https://doi.org/10.1007/s00031-008-9029-6