New realization of $\imath$quantum groups via $\Delta$-Hall algebras

Abstract

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