Quantum symmetric pairs via Hall algebras

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