Algebras and Representation Theory

, Volume 14, Issue 5, pp 897–948

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


    • Department of MathematicsUniversity of Oslo
  • Lars Tuset
    • Faculty of EngineeringOslo University College
Open AccessArticle

DOI: 10.1007/s10468-010-9223-9

Cite this article as:
Neshveyev, S. & Tuset, L. Algebr Represent Theor (2011) 14: 897. doi:10.1007/s10468-010-9223-9


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.


Quantum groups Drinfeld category Quasi-bialgebras Unitary twist

Mathematics Subject Classifications (2010)

Primary 17B37; Secondary 20G42 46L65

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© The Author(s) 2010