Integrable modules over quantum symmetric pair coideal subalgebras

We introduce the notion of integrable modules over $\mathrm{ı}$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\mathrm{ı}$quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\mathrm{ı}$quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\mathrm{ı}$quantum group.