Quantum symmetric pairs via Hall algebras

Abstract A quantum symmetric pair consists of a quantum group $\widetilde{\mathbf{U}}$ and its co-ideal 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 co-ideal 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. Dedicated to Professor Bin Zhu on the occasion of his 60th birthday