TY - JOUR
T1 - Quantization of lie bialgebras, Part VI
T2 - Quantization of generalized Kac-Moody algebras
AU - Etingof, Pavel
AU - Kazhdan, David
PY - 2008/12
Y1 - 2008/12
N2 - 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 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 g(A) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).
AB - 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 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 g(A) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).
UR - http://www.scopus.com/inward/record.url?scp=54449084451&partnerID=8YFLogxK
U2 - 10.1007/s00031-008-9029-6
DO - 10.1007/s00031-008-9029-6
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AN - SCOPUS:54449084451
SN - 1083-4362
VL - 13
SP - 527
EP - 539
JO - Transformation Groups
JF - Transformation Groups
IS - 3-4
ER -