Fused K-operators and the $q$-Onsager algebra
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914427723317248 |
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| author | Lemarthe, Guillaume Baseilhac, Pascal Gainutdinov, Azat M. |
| author_facet | Lemarthe, Guillaume Baseilhac, Pascal Gainutdinov, Azat M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00781 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fused K-operators and the $q$-Onsager algebra Lemarthe, Guillaume Baseilhac, Pascal Gainutdinov, Azat M. Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 81R50, 81R10, 81U15 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. |
| title | Fused K-operators and the $q$-Onsager algebra |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 81R50, 81R10, 81U15 |
| url | https://arxiv.org/abs/2301.00781 |