Quantum symmetric pairs via Hall algebras

Abstract

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

arXiv comment32 pages