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An equivalence of fusion categories. (English) Zbl 0860.17040

Geom. Funct. Anal. 6, No. 2, 249-267 (1996); erratum ibid. 23, No. 2, 810-811 (2013).
By G. Moore and N. Seiberg [Classical and conformal field theory, Commun. Math. Phys. 123, 177-254 (1989; Zbl 0694.53074)] it is known that the category \(\widetilde{\mathcal O}_\kappa\) of integrable representations of fixed (integral) positive level for the universal central extension of a loop algebra has the structure of a rigid braided tensor category.
From D. Kazhdan and G. Lusztig [Tensor structures arising from affine Lie algebras, I–IV, J. Am. Math. Soc. 6, 905-947, 949-1011 (1993; Zbl 0786.17017); J. Am. Math. Soc. 7, 335-381 (1994; Zbl 0802.17007); 383-453 (1994; Zbl 0802.17008)] in conjunction with S. Gelfand and D. Kazhdan [Invent. Math. 109, 595-617 (1992; Zbl 0784.18003)] and H. Andersen [Isr. Math. Conf. Proc. 7, 1-12 (1993; Zbl 0845.17018)], it is known how to put a structure of the same type on a suitable subquotient \(\widetilde{\mathcal O}_{-\kappa}\) of the category of highest weight modules of the same Lie algebra in fixed (integral) negative level.
In this work, the author proves that these two rigid braided tensor categories are isomorphic. The idea of proof is well explained by the author in the introduction: “It is known [see P. Deligne, Une description de catégorie tressée (inspire par Drinfeld) (unpublished) and D. Kazhdan and G. Lusztig (loc. cit.)] that the data of a tensor category is equivalent to the data of a) a certain collection of local systems on the moduli spaces of configurations on a projective line; b) a certain collection of isomorphisms between these local systems and their specializations along the boundary of these moduli spaces. The local systems arising from the category \(\widetilde{\mathcal O}_\kappa\) first appeared in the work of V. Knizhnik and A. Zamolodchikov [Nucl. Phys. B 247, 83-103 (1984; Zbl 0661.17020)]. Their explicit description was given in [B. Feigin, V. Schechtman and A. Varchenko, Commun. Math. Phys. 170, 219-247 (1995; Zbl 0842.17043)]. It turns out that exactly the same description fits the local systems arising from \(\widetilde{\mathcal O}_{-\kappa}\).”

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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References:

[1] H. Andersen, Quantum groups, invariants of 3-manifolds and semisimple tensor categories, Israel Math. Conf. Proc. 7 (1993), 1–12. · Zbl 0845.17018
[2] H. Andersen, Tensor products of quantized tilting modules, CMP 149 (1992), 149–159. · Zbl 0760.17004
[3] H. Andersen, J. Paradowski, Fusion categories arising from semisimple Lie algebras, CMP 169 (1995), 563–588. · Zbl 0827.17010
[4] A. Beilinson, B. Feigin, B. Mazur, Introduction to algebraic field theories on curves, preprint.
[5] P. Deligne, Une description de catégorie tressée (inspiré par Drinfeld), unpublished.
[6] B. Feigin, The semiinfinite homology of Kac-Moody and Virasoro Lie algebras, Russian Math. Surveys 39 (1984), 155–156. · Zbl 0574.17008 · doi:10.1070/RM1984v039n02ABEH003112
[7] B. Feigin, V. Schechtman, A. Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models, II, CMP 170 (1995), 219–247. · Zbl 0842.17043
[8] B. Feigin, A. Zelevinsky, Representations of contragredient Lie algebras and Macdonald identities, in ”Representations of Lie Groups and Lie Algebras” (A.A. Kirillov, ed.), Ak. Kiado, Budapest (1985), 25–77.
[9] S. Gelfand, D. Kazhdan, Examples of tensor categories, Inv. Math. 109 (1992), 595–617. · Zbl 0784.18003 · doi:10.1007/BF01232042
[10] S. Gelfand, D. Kazhdan, Invariants of three-dimensional manifolds, to appear. · Zbl 0870.57017
[11] J.C. Jantzen, Representations of Algebraic Groups, Pure and Applied Mathematics 131, Academic Press, 1987. · Zbl 0654.20039
[12] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993), 905–947. · Zbl 0786.17017 · doi:10.1090/S0894-0347-1993-99999-X
[13] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, II, J. Amer. Math. Soc. 6 (1993), 949–1011. · Zbl 0786.17017 · doi:10.1090/S0894-0347-1993-1186962-0
[14] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, III, J. Amer. Math. Soc. 7 (1994), 335–381. · Zbl 0802.17007 · doi:10.1090/S0894-0347-1994-1239506-X
[15] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, IV, J. Amer. Math. Soc. 7 (1994), 383–453. · Zbl 0802.17008 · doi:10.1090/S0894-0347-1994-1239507-1
[16] A. Kirillov, Jr., On inner product in modular tensor categories, I, preprint q-alg/950817, to appear in J. amer. Math. Soc.
[17] V. Knizhnik, A. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B 247 (1984), 83–103. · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[18] S. Kumar, Extension of the categoryO and a vanishing theorem for the Ext functor for Kac-Moody algebras, J. of Algebra 108 (1987), 472–491. · Zbl 0625.17009 · doi:10.1016/0021-8693(87)90111-6
[19] G. Lusztig, Monodromic systems on affine flag manifolds, Proc. R. Soc. Lond. A 445 (1994), 231–246. · Zbl 0829.17018 · doi:10.1098/rspa.1994.0058
[20] G. Moore, N. Seiberg, Classical and conformal field theory, CMP 123 (1989), 177–254. · Zbl 0694.53074
[21] C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223. · Zbl 0725.16011 · doi:10.1007/BF02571521
[22] A. Rocha-Caridi, N.R. Wallach, Projective modules over graded Lie algebras, Math. Z. 180 (1982), 151–177. · Zbl 0475.17002 · doi:10.1007/BF01318901
[23] A. Tsuchiya, K. Ueno, Y. Yamada, Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries, Advanced Studies in Pure Math. 19 (1989), 459–566. · Zbl 0696.17010
[24] E. Verlinde, Fusion rules and modular transformations in 2-d conformal field theory, Nucl. Phys. B 300 (1988), 360–376. · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7
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