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On an inner product in modular tensor categories

Author(s): Alexander A. Kirillov Jr.
Journal: J. Amer. Math. Soc. 9 (1996), 1135-1169.
MSC (1991): Primary 81R50, 05E35, 18D10; Secondary 57M99
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References:

[A]
Andersen, H. H., On tensor products of quantized tilting modules, Com. Math. Phys. 149 (1992), 149--159. MR 94b:17015

[AI]
Askey, R. and Ismail, M.E.H., A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhäuser, 1983, pp. 55--78. MR 87a:33015

[AP]
Andersen, H. H. and Paradowski, J., Fusion categories arising from semisimple Lie algebras, Com. Math. Phys 169 (1995), 563--588. CMP 95:11

[Ch]
Cherednik, I., Macdonald's evaluation conjectures and difference Fourier transform, preprint, May 1995, q-alg/9412016. CMP 96:02

[Dr1]
Drinfeld, V.G., Quantum groups, Proc. Int. Congr. Math., Berkeley, 1986, pp. 798--820. MR 89f:17017

[Dr2]
------, On almost cocommutative Hopf algebras, Leningrad Math.J. 1 (2) (1990), 321--342. MR 91b:16046

[EK1]
Etingof, P.I. and Kirillov, A.A., Jr, A unified representation-theoretic approach to special functions, Functional Anal. and its Applic. 28 (1) (1994), 91--94. MR 95h:33010

[EK2]
------, Macdonald's polynomials and representations of quantum groups, Math. Res. Let. 1 (1994), 279--296. CMP 95:04

[EK3]
------, Representation-theoretic proof of inner product and symmetry identities for Macdonald's polynomials, hep-th/9410169, to appear in Comp. Math. (1995).

[F]
Finkelberg, M., Fusion categories, Ph.D thesis, Harvard Univ. (1993), (to appear in GAFA).

[GK]
Gelfand, S. and Kazhdan, D., Examples of tensor categories, Invent. Math. 109 (1992), 595--617. MR 93m:20057

[Kac]
Kac, V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, 1990. MR 92k:17038

[Kas]
Kassel, C., Quantum groups, Springer, New York, 1995. CMP 95:09

[KL1]
Kazhdan, D. and Lusztig, G., Tensor structures arising from affine Lie algebras. I, J. of AMS 6 (1993), 905--947. MR 93m:17014

[KL2]
------, Tensor structures arising from affine Lie algebras. II, J. of AMS 6 (1993), 949--1011. MR 93m:17014

[KL3]
------, Tensor structures arising from affine Lie algebras. III, J. of AMS 7 (1994), 335--381. MR 94g:17048

[KL4]
------, Tensor structures arising from affine Lie algebras. IV, J. of AMS 7 (1994), 383--453. MR 94g:17049

[Ke]
Kerler, T., Mapping class group actions on quantum doubles, Comm. Math. Phys. 168 (1995), 353--388. CMP 95:10

[KR]
Kirillov, A.N. and Reshetikhin, N.Yu., q-Weyl group and a multiplicative formula for universal R-matrices, Comm. Math. Phys. 134 (1990), 421--431. MR 92c:17023

[L1]
Lusztig, G., Modular representations and quantum groups, Contemp. Math., vol. 82, AMS, Providence, 1989, pp. 52--77. MR 90a:16008

[L2]
------, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras, J. of AMS 3 (1990), 257--296. MR 91e:17009

[L3]
------, Quantum groups at roots of 1, Geom. Dedicat. 35 (1990), 89--113. MR 91j:17018

[L4]
------, On quantum groups, J. of Algebra 131 (1990), 466--475. MR 91j:17019

[L5]
------, Introduction to quantum groups, Birkhäuser, Boston, 1993. MR 94m:17016

[L6]
------, Monodromic systems on affine flag manifolds, Proc. R. Soc. Lond. A 445 (1994), 231--246. MR 95m:20049

[L7]
------, Canonical bases arising from quantized enveloping algebras. II, Progr. Theor. Phys. Suppl. 102 (1990), 175--201. MR 93g:17019

[LS]
Levendorskii, S.Z. and Soibelman, Ya. S., Some applications of quantum Weyl groups, Jour. Geom. Phys. 7 (2) (1990), 241--254. MR 92g:17016

[Lyu]
Lyubashenko, V., Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995), 279--327. CMP 95:10

[M1]
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.

[M2]
------, Orthogonal polynomials associated with root systems, preprint (1988).

[Mac]
MacLane, S., Categories for working mathematician, Graduate Texts in Mathematics, vol. 5, Springer--Verlag, New York, 1971. MR 50:7275

[MS1]
Moore, G., Seiberg, N., Classical and quantum conformal field theory, Com. Math. Phys. 123 (1989), 177--254. MR 90e:81216

[MS2]
------, Lectures on RCFT, Superstrings '89 (Proc. of the 1989 Trieste Spring School) (M. Green et al, eds.), World Sci., River Edge, NJ, 1990, pp. 1--129. MR 93m:81133a

[MS3]
------, Polynomial equations for rational conformal field theories, Phys. Let. B212 (1988), 451--460. MR 89m:81155

[MS4]
------, Naturality in conformal field theory, Nucl. Phys. B313 (1989), 16--40. MR 90f:81119

[RT1]
Reshetikhin, N. and Turaev, V., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1--26. MR 91c:57016

[RT2]
------, Invariants of 3-manifolds via link polynomials and quantum groups, Inv. Math. 103 (1991), 547-597. MR 92b:57024

[Tu]
Turaev, V., Quantum invariants of knots and 3-manifolds, W. de Gruyter, Berlin, 1994. MR 95k:57014

[Vaf]
Vafa, C., Towards classification of conformal theories, Phys. Lett. B 206 (1988), 421--426. MR 89k:81178

[We]
Wenzl, H., Quantum groups and subfactors of type B, C, and D, Comm. Math. Phys. 133 (1990), 383--433. MR 92k:17032


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Additional Information:

Alexander A. Kirillov Jr.
Affiliation: Department of Mathematics, Massachusetts Institite of Technology, Cambridge, Massachusetts 02139
Email: kirillov@math.mit.edu

DOI: 10.1090/S0894-0347-96-00210-X
PII: S 0894-0347(96)00210-X
Keywords: Modular tensor categories, quantum groups at roots of 1, Macdonald polynomials
Received by editor(s): October 12, 1995
Received by editor(s) in revised form: November 20, 1995
Dedicated: Dedicated to my father on his 60th birthday
Copyright of article: Copyright 1996, American Mathematical Society


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