On an inner product in modular tensor categories
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References
- Henning Haahr Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), no. 1, 149–159. MR 1182414, DOI 10.1007/BF02096627
- R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
- Andersen, H. H. and Paradowski, J., Fusion categories arising from semisimple Lie algebras, Com. Math. Phys 169 (1995), 563–588.
- Cherednik, I., Macdonald’s evaluation conjectures and difference Fourier transform, preprint, May 1995, q-alg/9412016.
- V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
- V. G. Drinfel′d, Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989), no. 2, 30–46 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 2, 321–342. MR 1025154
- A. A. Kirillov Jr. and P. I. Ètingof, On a unified representation-theoretic approach to the theory of special functions, Funktsional. Anal. i Prilozhen. 28 (1994), no. 1, 91–94 (Russian); English transl., Funct. Anal. Appl. 28 (1994), no. 1, 73–76. MR 1275729, DOI 10.1007/BF01079011
- —, Macdonald’s polynomials and representations of quantum groups, Math. Res. Let. 1 (1994), 279–296.
- —, Representation-theoretic proof of inner product and symmetry identities for Macdonald’s polynomials, hep-th/9410169, to appear in Comp. Math. (1995).
- Finkelberg, M., Fusion categories, Ph.D thesis, Harvard Univ. (1993), (to appear in GAFA).
- Sergei Gelfand and David Kazhdan, Examples of tensor categories, Invent. Math. 109 (1992), no. 3, 595–617. MR 1176207, DOI 10.1007/BF01232042
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Kassel, C., Quantum groups, Springer, New York, 1995.
- D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
- D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
- D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), no. 2, 335–381. MR 1239506, DOI 10.1090/S0894-0347-1994-1239506-X
- D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), no. 2, 383–453. MR 1239507, DOI 10.1090/S0894-0347-1994-1239507-1
- Kerler, T., Mapping class group actions on quantum doubles, Comm. Math. Phys. 168 (1995), 353–388.
- A. N. Kirillov and N. Reshetikhin, $q$-Weyl group and a multiplicative formula for universal $R$-matrices, Comm. Math. Phys. 134 (1990), no. 2, 421–431. MR 1081014, DOI 10.1007/BF02097710
- G. Lusztig, Modular representations and quantum groups, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 59–77. MR 982278, DOI 10.1090/conm/082/982278
- George Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296. MR 1013053, DOI 10.1090/S0894-0347-1990-1013053-9
- George Lusztig, Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), no. 1-3, 89–113. MR 1066560, DOI 10.1007/BF00147341
- George Lusztig, On quantum groups, J. Algebra 131 (1990), no. 2, 466–475. MR 1058558, DOI 10.1016/0021-8693(90)90187-S
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- George Lusztig, Monodromic systems on affine flag manifolds, Proc. Roy. Soc. London Ser. A 445 (1994), no. 1923, 231–246. MR 1276910, DOI 10.1098/rspa.1994.0058
- G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175–201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, DOI 10.1143/PTPS.102.175
- S. Z. Levendorskiĭ and Ya. S. Soĭbel′man, Some applications of the quantum Weyl groups, J. Geom. Phys. 7 (1990), no. 2, 241–254. MR 1120927, DOI 10.1016/0393-0440(90)90013-S
- Lyubashenko, V., Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995), 279–327.
- Macdonald, I.G., A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien (1988), 131-171.
- —, Orthogonal polynomials associated with root systems, preprint (1988).
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038, DOI 10.1007/BF01238857
- Gregory Moore and Nathan Seiberg, Lectures on RCFT, Superstrings ’89 (Trieste, 1989) World Sci. Publ., River Edge, NJ, 1990, pp. 1–129. MR 1159969
- Gregory Moore and Nathan Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988), no. 4, 451–460. MR 962600, DOI 10.1016/0370-2693(88)91796-0
- Gregory Moore and Nathan Seiberg, Naturality in conformal field theory, Nuclear Phys. B 313 (1989), no. 1, 16–40. MR 984288, DOI 10.1016/0550-3213(89)90511-7
- N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112, DOI 10.1007/BF02096491
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
- V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673, DOI 10.1515/9783110883275
- Cumrun Vafa, Toward classification of conformal theories, Phys. Lett. B 206 (1988), no. 3, 421–426. MR 944264, DOI 10.1016/0370-2693(88)91603-6
- Hans Wenzl, Quantum groups and subfactors of type $B$, $C$, and $D$, Comm. Math. Phys. 133 (1990), no. 2, 383–432. MR 1090432, DOI 10.1007/BF02097374
Additional Information
- Alexander A. Kirillov Jr.
- Affiliation: Department of Mathematics, Massachusetts Institite of Technology, Cambridge, Massachusetts 02139
- Email: kirillov@math.mit.edu
- Received by editor(s): October 12, 1995
- Received by editor(s) in revised form: November 20, 1995
- © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc. 9 (1996), 1135-1169
- MSC (1991): Primary 81R50, 05E35, 18D10; Secondary 57M99
- DOI: https://doi.org/10.1090/S0894-0347-96-00210-X
- MathSciNet review: 1358983
Dedicated: Dedicated to my father on his 60th birthday