Notes on the Kazhdan-Lusztig theorem on equivalence of the Drinfeld category and the category of Uqg-Modules

Abstract

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g,h) of g-modules and the category of finite dimensional Uqg-modules, q=eπih, for h ∈ ℂ\ℚ*. Aiming at operator algebraists the result is formulated as the existence for each h ∈ iℝ of a normalized unitary 2-cochain F on the dual Ǧ of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by F *-isomorphic to the convolution algebra of the q-deformation G q of G, while the coboundary of F-1 coincides with Drinfeld's KZ-associator defined via monodromy of the Knizhnik-Zamolodchikov equations. © 2010 The Author(s).