Preprojective algebras and MV polytopes

Baumann, Pierre; Kamnitzer, Joel

doi:10.1090/S1088-4165-2012-00413-7

Preprojective algebras and MV polytopes
HTML articles powered by AMS MathViewer

by Pierre Baumann and Joel Kamnitzer

Represent. Theory 16 (2012), 152-188

DOI: https://doi.org/10.1090/S1088-4165-2012-00413-7

Published electronically: March 12, 2012

PDF | Request permission

Abstract:

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig’s nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.

References

Jared E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3, 567–588. MR 1958098, DOI 10.1215/S0012-7094-03-11636-1
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
P. Baumann, J. Kamnitzer, P. Tingley, Affine Mirković-Vilonen polytopes, in preparation.
Arkady Berenstein and Andrei Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128–166. MR 1456321, DOI 10.1007/PL00000363
I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
A. B. Buan, O. Iyama, I. Reiten, and J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math. 145 (2009), no. 4, 1035–1079. MR 2521253, DOI 10.1112/S0010437X09003960
William Crawley-Boevey, On the exceptional fibres of Kleinian singularities, Amer. J. Math. 122 (2000), no. 5, 1027–1037. MR 1781930, DOI 10.1353/ajm.2000.0036
William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
Christof Geiss, Bernard Leclerc, and Jan Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 825–876 (English, with English and French summaries). MR 2427512, DOI 10.5802/aif.2371
C. Geiß, B. Leclerc, J. Schröer, Kac-Moody groups and cluster algebras, Adv. Math. 228 (2011), 329–433.
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal, Fundamentals of convex analysis, Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. Abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [ibid.; MR1295240 (95m:90002)]. MR 1865628, DOI 10.1007/978-3-642-56468-0
Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
Joel Kamnitzer, Mirković-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245–294. MR 2630039, DOI 10.4007/annals.2010.171.245
J. Kamnitzer, C. Sadanand, Modules with 1-dimensional socle and components of Lusztig quiver varieties in type A, preprint, arXiv:1009.0272.
Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9–36. MR 1458969, DOI 10.1215/S0012-7094-97-08902-X
Y. Kimura, Affine quivers and crystal bases, Master thesis, University of Kyoto (Japan), 2007.
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175–201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, DOI 10.1143/PTPS.102.175
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
George Lusztig, Braid group action and canonical bases, Adv. Math. 122 (1996), no. 2, 237–261. MR 1409422, DOI 10.1006/aima.1996.0061
G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141–182. MR 1623674, DOI 10.1006/aima.1998.1729
G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), no. 2, 129–139. MR 1758244, DOI 10.1006/aima.1999.1873
Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI 10.1215/S0012-7094-94-07613-8
Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI 10.1215/S0012-7094-98-09120-7
C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224 (German). MR 576602, DOI 10.1007/BF02566682
Claus Michael Ringel, PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. MR 1370206, DOI 10.1515/crll.1996.470.51
Claus Michael Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 467–480. MR 1648647
Yoshihisa Saito, PBW basis of quantized universal enveloping algebras, Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, 209–232. MR 1265471, DOI 10.2977/prims/1195166130
Yoshihisa Saito, Crystal bases and quiver varieties, Math. Ann. 324 (2002), no. 4, 675–688. MR 1942245, DOI 10.1007/s00208-002-0332-6
Y. Saito, personal communication, November 2010.
Alistair Savage, Geometric and combinatorial realizations of crystals of enveloping algebras, Lie algebras, vertex operator algebras and their applications, Contemp. Math., vol. 442, Amer. Math. Soc., Providence, RI, 2007, pp. 221–232. MR 2372565, DOI 10.1090/conm/442/08528