Skip to main content Accessibility help

Quivers with loops and generalized crystals

  • Tristan Bozec (a1)


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.



Hide All
[Boz15] Bozec, T., Quivers with loops and perverse sheaves , Math. Ann. 362 (2015), 773797; MR 3368082.
[HLR13] Hausel, T., Letellier, E. and Rodriguez-Villegas, F., Arithmetic harmonic analysis on character and quiver varieties II , Adv. Math. 234 (2013), 85128; MR 3003926.
[HR08] Hausel, T. and Rodriguez-Villegas, F., Mixed Hodge polynomials of character varieties , Invent. Math. 174 (2008), 555624; with an appendix by Nicholas M. Katz; MR 2453601 (2010b:14094).
[JKK05] Jeong, K., Kang, S.-J. and Kashiwara, M., Crystal bases for quantum generalized Kac–Moody algebras , Proc. Lond. Math. Soc. (3) 90 (2005), 395438; MR 2142133 (2006e:17020).
[Jos95] 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).
[KKS09] Kang, S.-J., Kashiwara, M. and Schiffmann, O., Geometric construction of crystal bases for quantum generalized Kac–Moody algebras , Adv. Math. 222 (2009), 9961015; MR 2553376 (2010h:17016).
[Kas91] Kashiwara, M., On crystal bases of the Q-analogue of universal enveloping algebras , Duke Math. J. 63 (1991), 465516; MR 1115118 (93b:17045).
[KS97] Kashiwara, M. and Saito, Y., Geometric construction of crystal bases , Duke Math. J. 89 (1997), 936; MR 1458969 (99e:17025).
[KS94] 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).
[Li] Li, Y., Canonical bases of Cartan–Borcherds type, II, Preprint.
[Lus91] Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras , J. Amer. Math. Soc. 4 (1991), 365421; MR 1088333 (91m:17018).
[Lus00] Lusztig, G., Semicanonical bases arising from enveloping algebras , Adv. Math. 151 (2000), 129139; MR 1758244 (2001e:17033).
[MO12] Maulik, D. and Okounkov, A., Quantum groups and quantum cohomology, Preprint (2012),arXiv:1211.1287.
[Nak94] Nakajima, H., Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras , Duke Math. J. 76 (1994), 365416.
[Nak98] Nakajima, H., Quiver varieties and Kac–Moody algebras , Duke Math. J. 91 (1998), 515560; MR 1604167 (99b:17033).
[Nak01] Nakajima, H., Quiver varieties and tensor products , Invent. Math. 146 (2001), 399449; MR 1865400 (2003e:17023).
[Nak09] Nakajima, H., Quiver varieties and branching , SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 003, 37 pp.; MR 2470410 (2010f:17034).
[Sai02] Saito, Y., Crystal bases and quiver varieties , Math. Ann. 324 (2002), 675688; MR 1942245 (2004a:17023).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO

Quivers with loops and generalized crystals

  • Tristan Bozec (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.