  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 $Q$.
