$\imath$Hall algebras and $\imath$quantum groups

Abstract

We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall algebras give us $\imath$Hall algebras. We then use these $\imath$Hall algebras to realize quasi-split $\imath$quantum groups arising from quantum symmetric pairs. Relative braid group symmetries on $\imath$quantum groups are realized via reflection functors. In case of Jordan $\imath$quiver, the $\imath$Hall algebra is commutative and connections to $\imath$Hall-Littlewood symmetric functions are developed. In case of $\imath$quivers of diagonal type, our construction amounts to a reformulation of Bridgeland-Hall algebra realization of the Drinfeld double quantum groups (which in turn generalizes Ringel-Hall algebra realization of halves of quantum groups). Many rank 1 and rank 2 computations are supplied to illustrate the general constructions. We also briefly review $\imath$Hall algebras of weighted projective lines, and use them to realize Drinfeld type presentations of $\imath$quantum loop algebras.