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Homological approach to the Hernandez-Leclerc construction and quiver varieties


Authors: Giovanni Cerulli Irelli, Evgeny Feigin and Markus Reineke
Journal: Represent. Theory 18 (2014), 1-14
MSC (2010): Primary 14L30, 14M15, 16G20, 18F99
DOI: https://doi.org/10.1090/S1088-4165-2014-00449-7
Published electronically: January 13, 2014
MathSciNet review: 3149614
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Abstract: In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients.


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  • [1] S. Abeasis, A. Del Fra, and H. Kraft, The geometry of representations of $ A_{m}$, Math. Ann. 256 (1981), no. 3, 401-418. MR 626958 (83h:14038), https://doi.org/10.1007/BF01679706
  • [2] Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, Techniques of representation theory. London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. MR 2197389 (2006j:16020)
  • [3] Klaus Bongartz, Algebras and quadratic forms, J. London Math. Soc. (2) 28 (1983), no. 3, 461-469. MR 724715 (85i:16036), https://doi.org/10.1112/jlms/s2-28.3.461
  • [4] Klaus Bongartz, A geometric version of the Morita equivalence, J. Algebra 139 (1991), no. 1, 159-171. MR 1106345 (92f:16008), https://doi.org/10.1016/0021-8693(91)90288-J
  • [5] Klaus Bongartz, Minimal singularities for representations of Dynkin quivers, Comment. Math. Helv. 69 (1994), no. 4, 575-611. MR 1303228 (96f:16016), https://doi.org/10.1007/BF02564505
  • [6] Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245-287. MR 1402728 (98e:16012), https://doi.org/10.1006/aima.1996.0053
  • [7] Giovanni Cerulli Irelli, Evgeny Feigin, and Markus Reineke, Quiver Grassmannians and degenerate flag varieties, Algebra Number Theory 6 (2012), no. 1, 165-194. MR 2950163, https://doi.org/10.2140/ant.2012.6.165
  • [8] G. Cerulli Irelli, E. Feigin, and M. Reineke, Desingularization of quiver Grassmannians for Dynkin quivers, Adv. Math. 245 (2013), 182-207. MR 3084427, https://doi.org/10.1016/j.aim.2013.05.024
  • [9] William Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), no. 1, 55-79. MR 1957264 (2004c:16017), https://doi.org/10.1007/s00208-002-0367-8
  • [10] Evgeny Feigin and Michael Finkelberg, Degenerate flag varieties of type A: Frobenius splitting and BW theorem, Math. Z. 275 (2013), no. 1-2, 55-77. MR 3101796, https://doi.org/10.1007/s00209-012-1122-9
  • [11] D. Hernandez, B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, Preprint 2011, arXiv:1109.0862
  • [12] A. Kirillov and J. Thind, Coxeter elements and periodic Auslander-Reiten quiver, J. Algebra 323 (2010), no. 5, 1241-1265. MR 2584955 (2011e:20060), https://doi.org/10.1016/j.jalgebra.2009.11.024
  • [13] G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141-182. MR 1623674 (2000c:16016), https://doi.org/10.1006/aima.1998.1729
  • [14] Lieven Le Bruyn and Claudio Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), no. 2, 585-598. MR 958897 (90e:16048), https://doi.org/10.2307/2001477
  • [15] B. Leclerc, P. Plamondon, Nakajima varieties and repetitive algebras, Preprint 2012, arXiv:1208.3910
  • [16] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)

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Additional Information

Giovanni Cerulli Irelli
Affiliation: Mathematisches Institut, Universität Bonn, Bonn, Germany 53115
Email: cerulli.math@googlemail.com

Evgeny Feigin
Affiliation: Department of Mathematics, National Research University Higher School of Economics, Russia, 117312, Moscow, Vavilova str. 7 – and – Tamm Department of Theoretical Physics, Lebedev Physics Institute, Russia
Email: evgfeig@gmail.com

Markus Reineke
Affiliation: Fachbereich C - Mathematik, Bergische Universität Wuppertal, D - 42097 Wuppertal, Germany
Email: reineke@math.uni-wuppertal.de

DOI: https://doi.org/10.1090/S1088-4165-2014-00449-7
Received by editor(s): March 13, 2013
Received by editor(s) in revised form: October 17, 2013
Published electronically: January 13, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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