Notes on the Kazhdan-Lusztig theorem on equivalence of the Drinfeld category and the category of Uq(g)-modules

TL;DR

This paper discusses the proof of the Kazhdan-Lusztig theorem, establishing an equivalence between the Drinfeld category of g-modules and the category of finite-dimensional Uq(g)-modules, with implications for operator algebras.

Contribution

It reformulates the Kazhdan-Lusztig theorem in operator algebra terms, showing the existence of a unitary 2-cochain linking the convolution algebras of G and its q-deformation G_q.

Findings

Existence of a normalized unitary 2-cochain F for each h in iR.

The convolution algebra of G with F-twisted coproduct is *-isomorphic to that of G_q.

The coboundary of F^{-1} matches Drinfeld's KZ-associator.

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 Uq(g)-modules, q=exp(\pi ih), for h\in C\Q*. Aiming at operator algebraists the result is formulated as the existence for each h\in iR of a normalized unitary 2-cochain F on the dual \hat G of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by F is *-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.