Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T07:57:09.327Z Has data issue: false hasContentIssue false
  • English
  • Français

On two cohomological Hall algebras

Part of: (Co)homology theory Lie algebras and Lie superalgebras Projective and enumerative geometry

Published online by Cambridge University Press:  29 January 2019

Yaping Yang  and
Gufang Zhao
Yaping Yang
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, ParkvilleVIC3010, Australia (yaping.yang1@unimelb.edu.au)
Gufang Zhao
Affiliation:
Department of Mathematics, University of Massachusetts, Amherst, MA01003, USA (gufangz@unimelb.edu.au)
Get access Rights & Permissions [Opens in a new window]

Abstract

We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced in [19] associated with each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot [15]. The other one is the critical CoHA defined by Kontsevich-Soibelman associated with each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients in the sheaf of vanishing cycles. In the present paper, we show that the critical CoHA, for the quiver with potential of Ginzburg, is isomorphic to the preprojective CoHA as algebras. As applications, we obtain an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

Keywords

Quiver varietyHall algebraYangianquiver with potentials

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1581 - 1607
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Current address: School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville VIC 3010, Australia.

References

1Behrend, K., Bryan, J. and Szendröi, B.. Motivic degree zero Donaldson-Thomas invariants. Invent. Math. 192 (2013), 111–160. MR3032328CrossRefGoogle Scholar
2Davison, B.. The critical COHA of a quiver with potential, preprint, (2015). Q. J. Math. 68 (2017), 635–703. arXiv:1311.7172.10.1093/qmath/haw053CrossRefGoogle Scholar
3Derksen, H., Weyman, J. and Zelevinsky, A.. Quivers with potentials and their representations. I. Mutations. Selecta Math. (N.S.) 14 (2008), 59–119. MR2480710.10.1007/s00029-008-0057-9CrossRefGoogle Scholar
4Dimca, A.. Sheaves in topology. Universitext. (Berlin: Springer-Verlag, 2004). xvi+236 pp.CrossRefGoogle Scholar
5Drinfeld, V. G.. Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798–820, Amer. Math. Soc., Providence, RI, 1987.Google Scholar
6Fulton, W.. Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2 (Berlin: Springer-Verlag, 1984), xi+470 pp. MR0732620.CrossRefGoogle Scholar
7Ginzburg, V.. Calabi-Yau algebras. Preprint, (2006). arXiv:0612139.Google Scholar
8Kato, K. and Saito, T.. Ramification theory for varieties over a perfect field. Ann. of Math. (2) 168 (2008), 33–96. MR2415398.10.4007/annals.2008.168.33CrossRefGoogle Scholar
9Keller, B.. Deformed Calabi-Yau completions. With an appendix by Michel Van den Bergh. J. Reine Angew. Math. 654 (2011), 125–180. MR2795754.Google Scholar
10Kontsevich, M. and Soibelman, Y.. Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Commun. Number Theory Phys. 5 (2011), 231–352. MR2851153.CrossRefGoogle Scholar
11Levine, M. and Morel, F.. Algebraic cobordism theory (Berlin: Springer, 2007). MR2286826.Google Scholar
12Massey, D.. The Sebastiani–Thom isomorphism in the Derived Category. Comp. Math. 125 (2001), 353–362.10.1023/A:1002608716514CrossRefGoogle Scholar
13Nakajima, H.. Quiver varieties and finite dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), 145–238. MR1808477 arXiv:9912158.CrossRefGoogle Scholar
14Ren, J. and Soibelman, Y.. Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 2-dimensional Calabi-Yau categories (with an appendix by Ben Davison). Algebra, geometry, and physics in the 21st century, Progr. Math., vol. 324, pp. 261–293 (Birkhäuser/Springer, Cham, 2017), arXiv:1508.06068.Google Scholar
15Schiffmann, O. and Vasserot, E.. The elliptic Hall algebra and the K-theory of the Hilbert scheme of ${\open A}^2$. Duke Math. J. 162 (2013), 279–366. MR3018956.10.1215/00127094-1961849CrossRefGoogle Scholar
16Soibelman, Y.. Remarks on Cohomological Hall algebras and their representations. Arbeitstagung Bonn 2013, Progr. Math., vol. 319, pp. 355–385, (Birkhäuser/Springer, Cham, 2016). arXiv:1404.1606.Google Scholar
17Szendröi, B.. Nekrasov's partition function and refined Donaldson-Thomas theory: the rank one case. SIGMA 8 (2012), 1–16. MR3007271.Google Scholar
18Varagnolo, M.. Quiver varieties and Yangians. Lett. Math. Phys. 53 (2000), 273–283. MR1818101.10.1023/A:1007674020905CrossRefGoogle Scholar
19Yang, Y. and Zhao, G.. The cohomological Hall algebra for a preprojective algebra. Proc. Lond. Math. Soc. 116 1029–1074, (May 2018). arXiv:1407.7994.CrossRefGoogle Scholar
20Yang, Y. and Zhao, G.. Cohomological Hall algebras and affine quantum groups. Selecta Math. 24 (2018), 1093–1119. arXiv:1604.01865.CrossRefGoogle Scholar
21Yang, Y. and Zhao, G.. Quiver varieties and elliptic quantum groups, preprint. arXiv:1708.01418.Google Scholar