Quantization of Lie bialgebras, part VI: quantization of generalized Kac-Moody algebras

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 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).

10 pages, amstex; the paper relies on the fact that quantization of Lie bialgebras commutes with duals and doubles, whose proof in the original version contained a gap. An adequate proof of this fact was obtained by Enriquez and Geer in 0707.2337. The present version takes this into account