Hall algebras and quantum symmetric pairs I: Foundations

A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\varsigma}$ with parameters $\varsigma$ (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 ${\mathbf U}^{\imath}_{\varsigma}$ is recovered by a central reduction of $\widetilde{\mathbf U}^{\imath}$. The modified 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 modified 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, and a reduced version provides a categorification of ${\mathbf U}^{\imath}_{\varsigma}$. Monomial bases and PBW bases for these Hall algebras and $\imath$quantum groups are constructed. In the special case of quivers of diagonal type, our construction reduces to a reformulation of Bridgeland's Hall algebra realization of quantum groups.