Quantum symmetric pairs via Hall algebras

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 $Δ:\widetilde{\mathbf{U}}^\imath\to \widetilde{\mathbf{U}}^\imath\otimes \widetilde{\mathbf{U}}$ preserve the integral forms of $\widetilde{\mathbf{U}}^\imath$ and $\widetilde{\mathbf{U}}$, which are used to construct the dual canonical bases.

💡 Research Summary

The paper develops a comprehensive Hall‑algebra framework that simultaneously realizes the quantum group (\widetilde{\mathbf U}) (the Drinfeld‑double version of the Drinfeld–Jimbo quantum group) and its coideal subalgebra (\widetilde{\mathbf U}^\imath) (the universal quasi‑split i‑quantum group) as a quantum symmetric pair. Building on Bridgeland’s construction of (\widetilde{\mathbf U}) via the Hall algebra of 2‑periodic complexes of projective modules, and on Lu–Wang’s i‑Hall algebra realization of (\widetilde{\mathbf U}^\imath) from i‑quiver algebras, the authors unify these two realizations using the theory of i‑Hopf algebras.

The main technical achievements are as follows:

1. Algebraic Isomorphisms. For a Dynkin i‑quiver ((Q,\tau)) there is an established isomorphism (\psi: \widetilde{\mathbf U}^\imath \to \mathcal H(Q,\tau)) (Lemma 3.5). For the diagonal i‑quiver ((Q_{\mathrm{dbl}},\mathrm{swap})) there is an isomorphism (\psi_{\mathrm{dbl}}: \widetilde{\mathbf U} \to \mathcal H(Q_{\mathrm{dbl}},\mathrm{swap})) (Bridgeland’s result). The extended Hall algebra (\mathcal B(Q)) (the classical Ringel–Hall algebra) embeds as a Hopf subalgebra of (\mathcal H(Q_{\mathrm{dbl}},\mathrm{swap})); its i‑Hopf counterpart (\mathcal B^\imath_\varrho(Q)) and the Drinfeld double (D(\mathcal B(Q))) are identified with (\mathcal H(Q,\tau)) and (\mathcal H(Q_{\mathrm{dbl}},\mathrm{swap})) via explicit algebra isomorphisms (\Phi) and (\Phi_{\mathrm{dbl}}) (Propositions 4.1, 4.3).

2. Embedding and Coideal Maps. The authors construct an injective algebra homomorphism \