Hall algebras and quantum symmetric pairs I: Foundations

Abstract

A quantum symmetric pair consists of a quantum group U $\mathbf {U}$ and its coideal subalgebra U ς ı ${\mathbf {U}}^{\imath }_{\bm{\varsigma }}$ with parameters ς $\bm{\varsigma }$ (called an ı $\imath$ quantum group). We initiate a Hall algebra approach for the categorification of ı $\imath$ quantum groups. A universal ı $\imath$ quantum group U ∼ ı $\widetilde{\mathbf {U}}^{\imath }$ is introduced and U ς ı ${\mathbf {U}}^{\imath }_{\bm{\varsigma }}$ is recovered by a central reduction of U ∼ ı $\widetilde{\mathbf {U}}^{\imath }$ . 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 the Appendix by the first author. A new class of 1-Gorenstein algebras (called ı $\imath$ quiver algebras) arising from acyclic quivers with involutions is introduced. The semi-derived Ringel–Hall algebras for the Dynkin ı $\imath$ quiver algebras are shown to be isomorphic to the universal quasi-split ı $\imath$ quantum groups of finite type. Monomial bases and PBW bases for these Hall algebras and ı $\imath$ quantum groups are constructed.