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

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 $H=\mathcal{L}{U}_q{\mathfrak{s}\mathfrak{l}}_2$, the quantum loop algebra for ${\mathfrak{s}\mathfrak{l}}_2$, and $B={\mathcal{A}}_q$, the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $\mathcal{L}{U}_q{\mathfrak{s}\mathfrak{l}}_2$ 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.