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.
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Bakalov, B., Kirillov, Jr., A.: Lectures on tensor categories and modular functors. In: University Lecture Series, vol. 21. American Mathematical Society, Providence, RI (2001)
Bichon, J.: The representation category of the quantum group of a non-degenerate bilinear form. Commun. Algebra 31, 4831–4851 (2003)
Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1995)
Drinfeld, V.G.: Quasi-Hopf algebras. Leningr. Math. J. 1, 1419–1457 (1990)
Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({\rm Gal}(\overline {\mathbb Q}/{\mathbb Q})\). Leningr. Math. J. 2, 829–860 (1991)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. III. Based on Notes Left by Harry Bateman, Robert E. Krieger Publishing Co., Inc., Melbourne, FL (1981)
Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras. I. Sel. Math. (N.S.) 2, 1–41 (1996)
Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras. VI. math.QA/0004042 (preprint)
Etingof, P., Ostrik, V.: Finite tensor categories. Mosc. Math. J. 4, 627–654, 782–783 (2004)
Häring-Oldenburg, R.: Reconstruction of weak quasi Hopf algebras. J. Algebra 194, 14–35 (1997)
Kassel, C.: Quantum Groups. Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. III. J. Am. Math. Soc. 7, 335–381 (1994)
Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. IV. J. Am. Math. Soc. 7, 383–453 (1994)
Le, T.T.Q., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial. Nagoya Math. J. 142, 39–65 (1996)
Lusztig, G.: Introduction to quantum groups. In: Progress in Mathematics, vol. 110. Birkhäuser Boston, Inc., Boston, MA (1993)
Majid, S.: Tannaka–Kreĭn theorem for quasi-Hopf algebras and other results. In: Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math., vol. 134, pp. 219–232. American Mathematical Society, Providence, RI (1992)
Neshveyev, S., Tuset, L.: The Dirac operator on compact quantum groups. J. Reine Angew. Math. 641, 1–20 (2010). arXiv:math/0703161v2 [math.OA]
Schauenburg, P.: Hopf algebra extensions and monoidal categories. In: New Directions in Hopf Algebras. Math. Sci. Res. Inst. Publ., vol. 43, pp. 321–381. Cambridge University Press, Cambridge (2002)
Schauenburg, P.: Two characterizations of finite quasi-Hopf algebras. J. Algebra 273, 538–550 (2004)
Ulbrich, K.-H.: On Hopf algebras and rigid monoidal categories. Isr. J. Math. 72, 252–256 (1990)
Van Daele, A.: Discrete quantum groups. J. Algebra 180, 431–444 (1996)
Wasow, W.: Asymptotic expansions for ordinary differential equations. Dover, New York (1987)
Woronowicz, S.L.: Tannaka–Kreĭn duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent. Math. 93, 35–76 (1988)
Zhu, Y.: Hecke algebras and representation ring of Hopf algebras. In: First International Congress of Chinese Mathematicians (Beijing, 1998). AMS/IP Stud. Adv. Math., vol. 20, pp. 219–227. American Mathematical Society, Providence, RI (2001)
Želobenko, D.P.: Compact Lie groups and their representations. In: Translations of Mathematical Monographs, vol. 40. American Mathematical Society, Providence, RI (1973)
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Neshveyev, S., Tuset, L. Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of \(\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}\)-Modules . Algebr Represent Theor 14, 897–948 (2011). https://doi.org/10.1007/s10468-010-9223-9
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DOI: https://doi.org/10.1007/s10468-010-9223-9