Quantum symmetric pairs via Hall algebras

Abstract

A quantum symmetric pair consists of a quantum group $\widetilde{\mathbf{U}}$ and its coideal subalgebra $\widetilde{\mathbf{U}}^\imath$. The Hall algebra constructions of $\widetilde{\mathbf{U}}$ and $\widetilde{\mathbf{U}}^\imath$ are given by Bridgeland and Lu--Wang, respectively. In this paper, we construct a Hall algebra framework for the coideal subalgebra structure of $\widetilde{\mathbf{U}}^\imath$ in $\widetilde{\mathbf{U}}$, and for the quantum symmetric pair $(\widetilde{\mathbf{U}},\widetilde{\mathbf{U}}^\imath)$. As an application, we prove that the natural embedding $\imath:\widetilde{\mathbf{U}}^\imath\to \widetilde{\mathbf{U}}$, and the coproduct $\Delta:\widetilde{\mathbf{U}}^\imath\to \widetilde{\mathbf{U}}^\imath\otimes \widetilde{\mathbf{U}}$ preserve the integral forms of $\widetilde{\mathbf{U}}^\imath$ and $\widetilde{\mathbf{U}}$, which are used to construct the dual canonical bases.