Journal article

On two cohomological Hall algebras

Yaping Yang, Gufang Zhao

Proceedings of the Royal Society of Edinburgh: Section A Mathematics | CAMBRIDGE UNIV PRESS | Published : 2020

Abstract

AbstractWe 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 po..

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Funding Acknowledgements

The authors thank Ben Davison for pointing out the aforementioned error in an earlier version of theorem 1.1, and for drawing our attention to the appendix of 14]. Most of the work was done when Y.Y. was hosted by MPIM and G.Z. was supported by Fondation Sciences Mathematiques de Paris and CNRS.