Geiß, Christof; Leclerc, Bernard; Schröer, Jan Quivers with relations for symmetrizable Cartan matrices. III: Convolution algebras. (English) Zbl 1362.16018 Represent. Theory 20, 375-413 (2016). 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)]. Cited in 4 ReviewsCited in 11 Documents 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 Keywords:enveloping algebra; convolution algebra; Iwanaga-Gorenstein algebras Citations:Zbl 1408.16012; Zbl 1362.16018 PDFBibTeX XMLCite \textit{C. Geiß} et al., Represent. Theory 20, 375--413 (2016; Zbl 1362.16018) Full Text: DOI arXiv References: [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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.