Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$ -Modules

Abstract

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category ${\mathcal D}({\mathfrak g},\hbar)$ of ${\mathfrak g}$ -modules and the category of finite dimensional $U_q{\mathfrak g}$ -modules, $q=e^{\pi i\hbar}$ , for $\hbar\in{\mathbb C}\setminus{\mathbb Q}^*$ . Aiming at operator algebraists the result is formulated as the existence for each $\hbar\in i{\mathbb R}$ of a normalized unitary 2-cochain ${\mathcal F}$ on the dual $\hat G$ of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by ${\mathcal F}$ is *-isomorphic to the convolution algebra of the q-deformation G q of G, while the coboundary of ${\mathcal F}^{-1}$ coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.