  A quantum symmetric pair consists of a quantum group $\mathbf U$ and its
coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbol{\varsigma}}$ with
parameters $\boldsymbol{\varsigma}$ (called an $\imath$quantum group). We
initiate a Hall algebra approach for the categorification of $\imath$quantum
groups. A universal $\imath$quantum group $\widetilde{\mathbf U}^{\imath}$ is
introduced and ${\mathbf U}^{\imath}_{\boldsymbol{\varsigma}}$ is recovered by
a central reduction of $\widetilde{\mathbf U}^{\imath}$. The semi-derived
Ringel-Hall algebras of the first author and Peng, which are closely related to
semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are
extended to the setting of 1-Gorenstein algebras, as shown in Appendix A by the
first author. A new class of 1-Gorenstein algebras (called $\imath$quiver
algebras) arising from acyclic quivers with involutions is introduced. The
semi-derived Ringel-Hall algebras for the Dynkin $\imath$quiver algebras are
shown to be isomorphic to the universal quasi-split $\imath$quantum groups of
finite type. Monomial bases and PBW bases for these Hall algebras and
$\imath$quantum groups are constructed.

Representation Theory