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Quivers with relations for symmetrizable Cartan matrices. III: Convolution algebras. (English) Zbl 1362.16018

Summary: We realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.
For Part II and III, see [Int. Math. Res. Not. 2018, No. 9, 2866–2898 (2018; Zbl 1408.16012); Represent. Theory 20, 375–413 (2016; Zbl 1362.16018)].

MSC:

16G20 Representations of quivers and partially ordered sets
17B35 Universal enveloping (super)algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16S30 Universal enveloping algebras of Lie algebras
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[1] Auslander, Maurice; Reiten, Idun; Smal{\o }, Sverre O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, xiv+425 pp. (1997), Cambridge University Press, Cambridge · Zbl 0834.16001
[2] Bia{\l }ynicki-Birula, A., On fixed point schemes of actions of multiplicative and additive groups, Topology, 12, 99-103 (1973) · Zbl 0255.14015
[3] Bongartz, Klaus, A geometric version of the Morita equivalence, J. Algebra, 139, 1, 159-171 (1991) · Zbl 0787.16011 · doi:10.1016/0021-8693(91)90288-J
[4] Bridgeland, Tom; Toledano Laredo, Valerio, Stability conditions and Stokes factors, Invent. Math., 187, 1, 61-98 (2012) · Zbl 1239.14008 · doi:10.1007/s00222-011-0329-4
[5] Crawley-Boevey, William; Schr{\"o}er, Jan, Irreducible components of varieties of modules, J. Reine Angew. Math., 553, 201-220 (2002) · Zbl 1062.16019 · doi:10.1515/crll.2002.100
[6] Dlab, Vlastimil; Ringel, Claus Michael, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc., 6, 173, v+57 pp. (1976) · Zbl 0332.16015
[7] C. Gei, B. Leclerc, J. Schr\"oer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Preprint (2014), 68 pp., arXiv:1410.1403.
[8] C. Gei, B. Leclerc, J. Schr\"oer, Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizers, Preprint (2015), 23 pp., arXiv:1511.05898.
[9] Joyce, Dominic, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math., 210, 2, 635-706 (2007) · Zbl 1119.14005 · doi:10.1016/j.aim.2006.07.006
[10] Kac, V. G., Infinite root systems, representations of graphs and invariant theory, Invent. Math., 56, 1, 57-92 (1980) · Zbl 0427.17001 · doi:10.1007/BF01403155
[11] Kac, Victor G., Infinite-dimensional Lie algebras, xxii+400 pp. (1990), Cambridge University Press, Cambridge · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[12] Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4, 2, 365-421 (1991) · Zbl 0738.17011 · doi:10.2307/2939279
[13] Lusztig, G., Semicanonical bases arising from enveloping algebras, Adv. Math., 151, 2, 129-139 (2000) · Zbl 0983.17009 · doi:10.1006/aima.1999.1873
[14] Riedtmann, Christine, Lie algebras generated by indecomposables, J. Algebra, 170, 2, 526-546 (1994) · Zbl 0841.16018 · doi:10.1006/jabr.1994.1351
[15] Ringel, Claus Michael, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics 1099, xiii+376 pp. (1984), Springer-Verlag, Berlin · Zbl 0546.16013 · doi:10.1007/BFb0072870
[16] Ringel, Claus Michael, Hall algebras and quantum groups, Invent. Math., 101, 3, 583-591 (1990) · Zbl 0735.16009 · doi:10.1007/BF01231516
[17] Ringel, Claus Michael, Lie algebras arising in representation theory. Representations of algebras and related topics, Kyoto, 1990, London Math. Soc. Lecture Note Ser. 168, 284-291 (1992), Cambridge Univ. Press, Cambridge · Zbl 0764.17005
[18] A. Schofield, Quivers and Kac-Moody Lie algebras, Unpublished manuscript, 23pp.
[19] Sweedler, Moss E., Hopf algebras, Mathematics Lecture Note Series, vii+336 pp. (1969), W. A. Benjamin, Inc., New York · Zbl 0194.32901
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