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Quivers with loops and generalized crystals

Part of: Representation theory of rings and algebras Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  20 July 2016

Tristan Bozec
Tristan Bozec*
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email tbozec@mit.edu
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Abstract

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In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

Keywords

quiver varietiesquantum groupsKashiwara crystals

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 1999 - 2040
Copyright
© The Author 2016 

References

Bozec, T., Quivers with loops and perverse sheaves , Math. Ann. 362 (2015), 773–797; MR 3368082.CrossRefGoogle Scholar
Hausel, T., Letellier, E. and Rodriguez-Villegas, F., Arithmetic harmonic analysis on character and quiver varieties II , Adv. Math. 234 (2013), 85–128; MR 3003926.CrossRefGoogle Scholar
Hausel, T. and Rodriguez-Villegas, F., Mixed Hodge polynomials of character varieties , Invent. Math. 174 (2008), 555–624; with an appendix by Nicholas M. Katz; MR 2453601 (2010b:14094).CrossRefGoogle Scholar
Jeong, K., Kang, S.-J. and Kashiwara, M., Crystal bases for quantum generalized Kac–Moody algebras , Proc. Lond. Math. Soc. (3) 90 (2005), 395–438; MR 2142133 (2006e:17020).CrossRefGoogle Scholar
Joseph, A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29 (Springer, Berlin, 1995); MR 1315966 (96d:17015).CrossRefGoogle Scholar
Kang, S.-J., Kashiwara, M. and Schiffmann, O., Geometric construction of crystal bases for quantum generalized Kac–Moody algebras , Adv. Math. 222 (2009), 996–1015; MR 2553376 (2010h:17016).CrossRefGoogle Scholar
Kashiwara, M., On crystal bases of the Q-analogue of universal enveloping algebras , Duke Math. J. 63 (1991), 465–516; MR 1115118 (93b:17045).CrossRefGoogle Scholar
Kashiwara, M. and Saito, Y., Geometric construction of crystal bases , Duke Math. J. 89 (1997), 9–36; MR 1458969 (99e:17025).CrossRefGoogle Scholar
Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292 (Springer, Berlin, 1994), with a chapter in French by Christian Houzel, corrected reprint of the 1990 original;MR 1299726 (95g:58222).Google Scholar
Li, Y., Canonical bases of Cartan–Borcherds type, II, Preprint.Google Scholar
Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras , J. Amer. Math. Soc. 4 (1991), 365–421; MR 1088333 (91m:17018).CrossRefGoogle Scholar
Lusztig, G., Semicanonical bases arising from enveloping algebras , Adv. Math. 151 (2000), 129–139; MR 1758244 (2001e:17033).CrossRefGoogle Scholar
Maulik, D. and Okounkov, A., Quantum groups and quantum cohomology, Preprint (2012),arXiv:1211.1287.Google Scholar
Nakajima, H., Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras , Duke Math. J. 76 (1994), 365–416.CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and Kac–Moody algebras , Duke Math. J. 91 (1998), 515–560; MR 1604167 (99b:17033).CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and tensor products , Invent. Math. 146 (2001), 399–449; MR 1865400 (2003e:17023).CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and branching , SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 003, 37 pp.; MR 2470410 (2010f:17034).Google Scholar
Saito, Y., Crystal bases and quiver varieties , Math. Ann. 324 (2002), 675–688; MR 1942245 (2004a:17023).CrossRefGoogle Scholar