On cohomological and K-theoretical Hall algebras of symmetric quivers

TL;DR

This paper explores the relationship between cohomological and K-theoretical Hall algebras of symmetric quivers, establishing algebra homomorphisms and category equivalences that connect their module categories.

Contribution

It demonstrates a homomorphism from K-theoretical to cohomological Hall algebra via Chern character and proves an equivalence of their locally finite module categories for symmetric quivers.

Findings

Existence of a homomorphism from KHA to a Zhang twist of CoHA.

Equivalence of categories of locally finite modules for the two algebras.

Examples of modules arising from cohomology and K-theory of quiver moduli spaces.

Abstract

We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^{\sigma}$ where $\hat{\mathcal{H}}^{\sigma}$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^{\sigma}-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat{\mathcal{H}}^{\sigma}$-, resp. $\mathcal{R}_{\mathbb Q}$- modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.