# Braided module categories via quantum symmetric pairs

## TL;DR

This paper explores the structure of quantum symmetric pair coideal subalgebras within quantum groups, establishing their braided module category structure and analyzing their centers using universal K-matrices.

## Contribution

It introduces a braided module category framework for quantum symmetric pair coideal subalgebras and describes their centers via universal K-matrices.

## Key findings

- Category of finite dimensional representations forms a braided module category.
- Universal K-matrix realizes the braiding for $B_{c,s}$.
- Describes a distinguished basis of the center of $B_{c,s}$.

## Abstract

Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the category of finite dimensional representations of a quantum symmetric pair coideal subalgebra $B_{c,s}$ of $U_q({\mathfrak g})$ is a braided module category over an equivariantization of $O_{int}$. The braiding for $B_{c,s}$ is realized by a universal K-matrix which lies in a completion of $B_{c,s}\otimes U_q({\mathfrak g})$. We apply these results to describe a distinguished basis of the center of $B_{c,s}$.