Universal K-matrix for quantum symmetric pairs

TL;DR

This paper constructs a universal K-matrix for quantum symmetric pair coideal subalgebras of quantized Kac-Moody algebras of finite type, extending the theory of universal R-matrices and connecting to ribbon category representations.

Contribution

It introduces the first construction of a universal K-matrix for quantum symmetric pairs in the finite type case, generalizing the universal R-matrix framework.

Findings

Universal K-matrix exists for quantum symmetric pairs of finite type.

Construction parallels the universal R-matrix for quantum groups.

Provides examples for ribbon category representation programs.

Abstract

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $U_q(\mathfrak{g})$ denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras $B_{c,s}$ of $U_q(\mathfrak{g})$ have a universal K-matrix if $\mathfrak{g}$ is of finite type. By a universal K-matrix for $B_{c,s}$ we mean an element in a completion of $U_q(\mathfrak{g})$ which commutes with $B_{c,s}$ and provides solutions of the reflection equation in all integrable $U_q(\mathfrak{g})$-modules in category $\mathcal{O}$. The construction of the universal K-matrix for $B_{c,s}$ bears significant resemblance to the construction of the universal R-matrix for $U_q(\mathfrak{g})$. Most steps in the construction of the universal K-matrix are performed in the general Kac-Moody setting. In the late nineties T. tom Dieck and R. H\"aring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.