On two cohomological Hall algebras

TL;DR

This paper demonstrates the isomorphism between two types of cohomological Hall algebras (CoHA) associated with quivers, linking preprojective and critical CoHAs, and explores their relation to Yangians.

Contribution

It establishes an isomorphism between the preprojective and critical CoHAs for Ginzburg quivers, connecting different algebraic frameworks in representation theory.

Findings

Proves the isomorphism between the two CoHAs.

Constructs an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

Generalizes known algebraic structures in quiver representation theory.

Abstract

We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced by the authors in arXiv:1407.7994 associated to 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. The other one is the critical CoHA defined by Kontsevich-Soibelman associated to 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.