On cohomological and K-theoretical Hall algebras of symmetric quivers

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 ${\hat{\mathcal{H}}}^{\sigma }-{\mathrm{Mod}}_{lf}\simeq {\mathcal{R}}_{\mathbb{Q}}-{\mathrm{Mod}}_{lf}$. 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.