, Volume 14, Issue 5, pp 897-948,
Open Access This content is freely available online to anyone, anywhere at any time.

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.

Presented by Alain Verschoren.