Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 026, 74 pages      arXiv:2301.00781      https://doi.org/10.3842/SIGMA.2026.026

Fused K-Operators and the $q$-Onsager Algebra

Guillaume Lemarthe, Pascal Baseilhac and Azat M. Gainutdinov
Institut Denis-Poisson CNRS/UMR 7013, Université de Tours, Université d'Orléans, Parc de Grammont, 37200 Tours, France

Received December 10, 2024, in final form February 12, 2026; Published online March 20, 2026

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.

Key words: reflection equation; universal K-matrix; $q$-Onsager algebra; fusion for K-operators.

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