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On two cohomological Hall algebras

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

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Footnotes

*

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

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