Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T11:31:38.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum groups via cyclic quiver varieties I

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

Published online by Cambridge University Press:  07 September 2015

Fan Qin
Fan Qin*
Affiliation:
Institut de Recherche Mathématique Avancée (IRMA), 7 rue René Descartes, 67084 Strasbourg Cedex, France email qin@math.unistra.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

Keywords

quantum groupquiver varietycategorificationdual canonical basis

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 2 , February 2016 , pp. 299 - 326
Copyright
© The Author 2015 

References

Bridgeland, T., Quantum groups via Hall algebras of complexes, Ann. of Math. (2) 177 (2013), 739–759.CrossRefGoogle Scholar
Fang, X. and Rosso, M., Multi-brace cotensor Hopf algebras and quantum groups, Preprint (2012), arXiv:1210.3096.Google Scholar
Geiß, C., Leclerc, B. and Schröer, J., Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), 337–397.CrossRefGoogle Scholar
Gorsky, M., Semi-derived Hall algebras and tilting invariance of Bridgeland–Hall algebras, Preprint (2013), arXiv:1303.5879.Google Scholar
Hernandez, D., Algebraic approach to q, t-characters, Adv. Math. 187 (2004), 1–52, doi:10.1016/j.aim.2003.07.016.CrossRefGoogle Scholar
Hernandez, D., The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond, J. Algebra 279 (2004), 514–547.CrossRefGoogle Scholar
Hernandez, D. and Leclerc, B., Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math. 701 (2015), 77–126, doi:10.1515/crelle-2013-0020.CrossRefGoogle Scholar
Kapranov, M., Heisenberg doubles and derived categories, J. Algebra 202 (1998), 712–744.CrossRefGoogle Scholar
Kashiwara, M., On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516, doi:10.1215/S0012-7094-91-06321-0;  MR 1115118 (93b:17045).CrossRefGoogle Scholar
Keller, B. and Scherotzke, S., Graded quiver varieties and derived categories, J. Reine Angew. Math. (2013), doi:10.1515/crelle-2013-0124.Google Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347.CrossRefGoogle Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups III, Quantum Topol. 1 (2010), 1–92.CrossRefGoogle Scholar
Kimura, Y., Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52 (2012), 277–331, doi:10.1215/21562261-1550976.CrossRefGoogle Scholar
Kimura, Y. and Qin, F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261–312.CrossRefGoogle Scholar
Leclerc, B., Dual canonical bases, quantum shuffles and q-characters, Math. Z. 246 (2004), 691–732.CrossRefGoogle Scholar
Leclerc, B. and Plamondon, P.-G., Nakajima varieties and repetitive algebras, Publ. RIMS Kyoto Univ. 49 (2013), 531–561.CrossRefGoogle Scholar
Lusztig, G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498.CrossRefGoogle Scholar
Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365–421.CrossRefGoogle Scholar
Lusztig, G., Introduction to quantum groups, Progress in Mathematics, vol. 110 (Birkhäuser, Boston, MA, 1993); MR 1227098 (94m:17016).Google Scholar
Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145–238; (electronic).CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), 1057–1097.CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), 71–126.CrossRefGoogle Scholar
Peng, L. and Xiao, J., Root categories and simple Lie algebras, J. Algebra 198 (1997), 19–56.CrossRefGoogle Scholar
Peng, L. and Xiao, J., Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (2000), 563–603.CrossRefGoogle Scholar
Qin, F., t-analog of q-characters, bases of quantum cluster algebras, and a correction technique, Int. Math. Res. Not. IMRN 2014 (2014), 6175–6232, doi:10.1093/imrn/rnt115.CrossRefGoogle Scholar
Ringel, C. M., Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–591.CrossRefGoogle Scholar
Rouquier, R., 2-Kac–Moody algebras, Preprint (2008), arXiv:0812.5023.Google Scholar
Rouquier, R., Lectures on canonical and crystal bases of Hall algebras, Preprint (2009),arXiv:0910.4460.Google Scholar
Schiffmann, O., Lectures on Hall algebras, Preprint (2006), arXiv:math.RT/0611617.Google Scholar
Varagnolo, M. and Vasserot, E., Perverse sheaves and quantum Grothendieck rings, in Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progress in Mathematics, vol. 210 (Birkhäuser, Boston, MA, 2003), 345–365; MR 1985732 (2004d:17023).CrossRefGoogle Scholar
Varagnolo, M. and Vasserot, E., Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67–100, doi:10.1515/crelle.2011.068.Google Scholar
Webster, B., Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products, Preprint (2010), arXiv:1001.2020.Google Scholar
Webster, B., Knot invariants and higher representation theory, Preprint (2013),arXiv:1309.3796.Google Scholar
Xiao, J., Xu, F. and Zhang, G., Derived categories and Lie algebras, Preprint (2006),arXiv:math/0604564.Google Scholar