New realization of $ı$quantum groups via $\hat{I}”$-Hall algebras

Abstract

For an essentially small hereditary abelian category $\mathcal{A}$, we define a new kind of algebra ${\mathcal{H}}_{\hat{I}}”(\mathcal{A})$, called the $\hat{I}”$-Hall algebra of $\mathcal{A}$. The basis of ${\mathcal{H}}_{\hat{I}}”(\mathcal{A})$ is the isomorphism classes of objects in $\mathcal{A}$, and the $\hat{I}”$-Hall numbers calculate certain three-cycles of exact sequences in $\mathcal{A}$. We show that the $\hat{I}”$-Hall algebra ${\mathcal{H}}_{\hat{I}}”(\mathcal{A})$ is isomorphic to the 1-periodic derived Hall algebra of $\mathcal{A}$. By taking suitable extension and twisting, we can obtain the $ı$Hall algebra and the semi-derived Hall algebra associated to $\mathcal{A}$ respectively. When applied to the the nilpotent representation category $\mathcal{A}=\mathrm{re}{\mathrm{p}}^{\mathrm{nil}}(\mathbf{k}Q)$ for an arbitrary quiver $Q$ without loops, the (\emph{resp.} extended) $\hat{I}”$-Hall algebra provides a new realization of the (\emph{resp.} universal) $ı$quantum group associated to $Q$.

19 pages