Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras

Geiß, Christof; Leclerc, Bernard; Schröer, Jan

doi:10.1090/ert/487

Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras
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by Christof Geiß, Bernard Leclerc and Jan Schröer

Represent. Theory 20 (2016), 375-413

DOI: https://doi.org/10.1090/ert/487

Published electronically: October 7, 2016

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Abstract:

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.

References

Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
A. Białynicki-Birula, On fixed point schemes of actions of multiplicative and additive groups, Topology 12 (1973), 99–103. MR 313261, DOI 10.1016/0040-9383(73)90024-4
Klaus Bongartz, A geometric version of the Morita equivalence, J. Algebra 139 (1991), no. 1, 159–171. MR 1106345, DOI 10.1016/0021-8693(91)90288-J
Tom Bridgeland and Valerio Toledano Laredo, Stability conditions and Stokes factors, Invent. Math. 187 (2012), no. 1, 61–98. MR 2874935, DOI 10.1007/s00222-011-0329-4
William Crawley-Boevey and Jan Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201–220. MR 1944812, DOI 10.1515/crll.2002.100
Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. MR 447344, DOI 10.1090/memo/0173
C. Geiß, B. Leclerc, J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Preprint (2014), 68 pp., arXiv:1410.1403.
C. Geiß, B. Leclerc, J. Schröer, Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizers, Preprint (2015), 23 pp., arXiv:1511.05898.
Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635–706. MR 2303235, DOI 10.1016/j.aim.2006.07.006
V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92. MR 557581, DOI 10.1007/BF01403155
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), no. 2, 129–139. MR 1758244, DOI 10.1006/aima.1999.1873
Christine Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), no. 2, 526–546. MR 1302854, DOI 10.1006/jabr.1994.1351
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
Claus Michael Ringel, Lie algebras arising in representation theory, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 284–291. MR 1211484
A. Schofield, Quivers and Kac-Moody Lie algebras, Unpublished manuscript, 23pp.
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485