Fused K-operators and the $q$-Onsager algebra

TL;DR

This paper develops a universal framework for K-operators related to reflection equations in quantum algebra, introducing fused K-operators for the $q$-Onsager algebra and analyzing their properties and explicit forms.

Contribution

It extends the universal K-matrix approach to include fused K-operators for the $q$-Onsager algebra, providing explicit formulas and conjectures on their relations.

Findings

Fused K-operators satisfy spectral-parameter dependent reflection equations.

Explicit expressions for small spin-$j$ K-operators are provided.

Conjectured relations between fused K-operators and universal K-matrices.

Abstract

We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra $H$, its comodule algebra $B$ and a pair of consistent twists. In our setting, the universal K-matrix is an element of $B\otimes H$ satisfying certain axioms, and we consider the case $H=\mathcal{L} U_q \mathfrak{sl}_2$, the quantum loop algebra for $\mathfrak{sl}_2$, and $B=\mathcal{A}_q$, the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $\mathcal{L} U_q \mathfrak{sl}_2$ in ''non-semisimple'' cases, the new set of axioms allows us to introduce and study fused K-operators of spin-$j$; in particular, to prove that for all $j\in\frac{1}{2}\mathbb{N}$ they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra ${\mathcal A}_q$ for small values of spin-$j$. The precise relation between the fused K-operators of spin-$j$ and evaluations of a universal K-matrix for ${\mathcal A}_q$ is conjectured based on supporting evidence. We finally discuss implications of our results on the K-operators for quantum integrable systems.