This is joint work with Jiayi Chen and   Yanan Lin.

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. 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 $Q$.