Integrable modules over quantum symmetric pair coideal subalgebras

TL;DR

This paper defines integrable modules over quantum symmetric pair coideal subalgebras, establishes their properties, and connects their matrix coefficients to Bao-Song's coordinate ring, advancing the understanding of quantum symmetric pairs.

Contribution

It introduces the concept of integrable modules over $ extit{ extbf{}}$quantum groups and proves their properties, linking them to coordinate rings of $ extit{ extbf{}}$quantum groups.

Findings

Every integrable module over a quantum group remains integrable when restricted to an $ extit{ extbf{}}$quantum group.

The space of matrix coefficients of simple integrable modules matches Bao-Song's coordinate ring.

Presented a new framework for integrable modules over quantum symmetric pairs.

Abstract

We introduce the notion of integrable modules over $\imath$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 $\imath$quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath$quantum group of finite type with specific parameters coincides with Bao-Song's coordinate ring of the $\imath$quantum group.