×

zbMATH — the first resource for mathematics

Fusion categories arising from semisimple Lie algebras. (English) Zbl 0827.17010
There are some interesting categories for a given finite Cartan datum, for example, the category \({\mathcal O}\) of certain finitely generated modules of the corresponding semisimple Lie algebra \({\mathfrak g}\), the category of rational modules of the corresponding semisimple, simply connected algebraic group \(G\) over a field of positive characteristic, the category of locally finite modules of the associated quantum algebra \(U\) specialized at an \(l\)-th root of unity. In this paper, the authors investigate a certain semisimple subcategory equipped with a ‘reduced’ tensor product for the three above mentioned situations respectively. Also, they determine the fusion rules for this tensor product in each case via known character formulas for the involved modules.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05 Representation theory for linear algebraic groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [A1] Andersen, H.H.: Tensor products of Quantized Tilting Modules. Commun. Math. Phys.149, 149–159 (1992) · Zbl 0760.17004 · doi:10.1007/BF02096627
[2] [A2] Andersen, H.H.: A New Proof of Old Character Formulas. Contemp. Math.88, 198–207 (1989)
[3] [APW] Andersen, H.H., Polo, P., Wen, K.: Representations of Quantum Algebras. Invent. Math.104, 1–59 (1991) · Zbl 0724.17012 · doi:10.1007/BF01245066
[4] [AW] Andersen, H.H., Wen, K.: Representations of Quantum Algebras. The Mixed Case. J. Reine Angew. Math.427, 35–50 (1992) · Zbl 0771.17010 · doi:10.1515/crll.1992.427.35
[5] [BB] Beilinson, A., Bernstein, J.: Localisation de g-modules. C.R. Acad. Sci.292, 15–18 (1981) · Zbl 0476.14019
[6] [BK] Brylinski, J.-L., Kashiwara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math.64, 387–410 (1981) · Zbl 0473.22009 · doi:10.1007/BF01389272
[7] [CI] Collingwood, D.H., Irving, R.: A decomposition theorem for certain self-dual modules in the categoryO. Duke Math. J.58, 89–102 (1989) · Zbl 0673.17003 · doi:10.1215/S0012-7094-89-05806-7
[8] [D] Donkin, S.: On tilting modules for algebraic groups. Math. Z.212, 39–60 (1993) · Zbl 0798.20035 · doi:10.1007/BF02571640
[9] [F] Finkelberg, M.: Fusion Categories. Ph.D. thesis, Harvard 1993 · Zbl 0860.17040
[10] [GM1] Georgiev, G., Mathieu, O.: Catégorie de fusion pour les groupes de Chevalley. C.R. Acad. Sci.315, 659–662 (1992) · Zbl 0801.20014
[11] [GM2] Georgiev, G., Mathieu, O.: Fusion rings for modular representations of Chevalley groups. Preprint · Zbl 0830.20064
[12] [H] Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0254.17004
[13] [I] Irving, R.: A filtered categoryO S and application. Mem. of the AMS419 (1990)
[14] [J1] Jantzen, J.C.: Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics750 · Zbl 0426.17001
[15] [J2] Jantzen, J.C.: Representations of Algebraic Groups. Academic Press 1987
[16] [KL1] Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math.53, 165–184 (1979) · Zbl 0499.20035 · doi:10.1007/BF01390031
[17] [KL2] Kazhdan, D., Lusztig, G.: Tensor Structures Arising From Affine Lie Algebras I–IV, preprints (1992–1993)
[18] [L1] Lusztig, G.: Some problems in the representation theory of finite Chevalley groups. Proc. Symp. Pure Math.37, 313–317 (1980)
[19] [L2] Lusztig, G.: Introduction to Quantum Groups. Birkhäuser 1993 · Zbl 0788.17010
[20] [L3] Lusztig, G.: Quantum Groups at Roots of 1. Geom. Ded.35, 89–114 (1993)
[21] [L4] Lusztig, G.: Finite dimensional Hopf algebras arising from quantized enveloping algebras. J. Am. Math. Soc.82, 257–296 (1990) · Zbl 0695.16006
[22] [PW] Parshall, B., Wang, J.-P.: Quantum Linear Groups. Mem. of the AMS439 (1991)
[23] [R] Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z.208, 209–223 (1991) · Zbl 0725.16011 · doi:10.1007/BF02571521
[24] [T] Thams, L.: Two classical results in the quantum mixed case. J. Reine Angew. Math.436, 129–153 (1993) · Zbl 0760.17015 · doi:10.1515/crll.1993.436.129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.