Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



$C^{*}$ tensor categories from quantum groups

Author: Hans Wenzl
Journal: J. Amer. Math. Soc. 11 (1998), 261-282
MSC (1991): Primary 81R50, 46L37
MathSciNet review: 1470857
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathfrak g$ be a semisimple Lie algebra and let $d$ be the ratio between the square of the lengths of a long and a short root. Moreover, let $\mathcal F$ be the quotient category of the category of tilting modules of $U_q\mathfrak g$ modulo the ideal of tilting modules with zero $q$-dimension for $q=e^{\pm\pi i/dl}$. We show that for $l$ a sufficiently large integer, the morphisms of $\mathcal F$ are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II$_1$ factor for each object of $\mathcal F$.

References [Enhancements On Off] (What's this?)

  • [A] H.H. Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1991), 149-159. MR 94b:17015
  • [AP] H.H. Andersen, J. Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), 563-588. MR 96e:17026
  • [D1] V. Drinfeld, Quantum groups, Proceedings ICM 1986, 798-820. MR 89f:17017
  • [D2] V. Drinfeld, On almost cocommutaive Hopf algebras, Leningrad Math. J. 1 (1990), 321-342. MR 91b:16046
  • [EK] D. Evans and Y. Kawahigashi, Orbifold subfactors from Hecke algebras, Comm. Math. Phys. 165 (1994), 445-484. MR 96c:46059
  • [GHJ] F. Goodman, P. de la Harpe and V.F.R. Jones, Coxeter graphs and towers of algebras, Springer Verlag, MSRI Publications 1989. MR 91c:46082
  • [GW] F.Goodman, H. Wenzl, Littlewood Richardson coefficients for Hecke algebras at roots of unity, Adv. Math. 82 (1990), 244-265. MR 91i:20013
  • [H] J. Humphreys, Introduction to Lie algebras and representation theory, Springer 1972. MR 48:2197
  • [J] V.F.R. Jones, Index for subfactors, Invent. Math 72 (1983), 1-25. MR 84d:46097
  • [JS] A. Joyal and R. Street, The Geometry of Tensor Calculus, I. Adv. Math. 88, n. 1 (1991). MR 92d:18011
  • [Kc] V. Kac, Infinite dimensional Lie algebras, Cambridge University Press, 1990. MR 92k:17038
  • [Ka] M. Kashiwara, On crystal bases of the $q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516. MR 93b:17045
  • [Ks] Ch. Kassel, Quantum groups, Springer, 1995. MR 96e:17041
  • [KL] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I-IV, J. of AMS 6/7 (1993). MR 93m:17014; MR 94g:17048; MR 94g:17049
  • [Ki] A. Kirillov, Jr., On an inner product in modular categories, J. of AMS 9 (1996), 1135-1170. MR 97f:18007
  • [KR] A.N. Kirillov and N.Yu. Reshetikhin, $q$-Weyl group and a multiplicative formula for universal $R$-matrices, Comm. Math. Phys. 134 (1990), 413-419. MR 92c:17023
  • [LS] S.Z. Levendorskii and Ya.S. Soibelman, Some applications of quantum Weyl groups, J. Geom. Phys. 7 (1990), 241-254. MR 92g:17016
  • [LR] R. Longo, J.E. Roberts, A theory of dimension, $K$-Theory, (to appear).
  • [Li] P. Littelmann, Paths and root operators in representation theory, Ann. Math. 142 (1995), 499-525. MR 96m:17011
  • [Lu] G. Lusztig, Introduction to quantum groups, Birkhäuser, 1993. MR 94m:17016
  • [OV] A.L. Onishchik, E.B. Vinberg, Lie groups and algebraic groups, Springer, 1990. MR 91g:22001
  • [P] M. Pimsner, A class of Markov traces, preprint.
  • [PP] M. Pimsner, S. Popa, Entropy and index for subfactors, Ann. scient. Ec. Norm. Sup., 19 (1986), 57-106. MR 87m:46120
  • [Po1] S. Popa, Classification of subfactors: reduction to commuting square, Invent. Math. 101 (1990), 19-43. MR 91h:46109
  • [Po2] S. Popa, An axiomatization of the lattice of higher relative commutants, Invent. Math. 120 (1995), 427-445. MR 96g:46051
  • [TW1] V. G. Turaev, H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, Int. J. Math. 4 (1993), 323-358. MR 94i:57019
  • [TW2] V. G. Turaev, H. Wenzl, Semisimple and modular tensor categories from link invariants, Math. Annalen 309 (1997), 411-461.
  • [T] V.G. Turaev, Quantum Invariants of Knots and $3$-Manifolds, de Gruyter, 1994. MR 95k:57014
  • [V] E. Verlinde, Fusion rules and modular transformation in 2D conformal field theory, Nuclear Physics B 300 (1988), 360-376. MR 89h:81238
  • [Wa1] A. Wassermann, Coactions and Yang-Baxter equations for ergodic actions and subfactors, LMS Lecture Notes 136 (1988), 203-236. MR 92d:46167
  • [Wa2] A. Wassermann, Operator algebras and conformal field theory, Proc. I.C.M. Zürich (1994), Birkhäuser. MR 97e:81143
  • [Wa3] A. Wassermann, Operator algebras and conformal field theory III (to appear).
  • [W1] H. Wenzl, Hecke algebras of type $A_{n}$ and subfactors. Invent. Math. 92 (1988), 349-383. MR 90b:46118
  • [W2] H. Wenzl, Quantum groups and subfactors of Lie type B, C and D, Comm. Math. Phys. 133 (1990), 383-433. MR 92k:17032
  • [X] F. Xu, Standard $\lambda $-lattices from quantum groups, preprint.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 81R50, 46L37

Retrieve articles in all journals with MSC (1991): 81R50, 46L37

Additional Information

Hans Wenzl
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Keywords: Quantum groups at roots of 1, modular tensor categories, subfactors
Received by editor(s): February 7, 1997
Received by editor(s) in revised form: September 29, 1997
Additional Notes: The author was supported in part by NSF grant # DMS 94-00987
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society