Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation
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arXiv
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| Natura: | Preprint |
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2025
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| author | Jedlička, Přemysl Pilitowska, Agata |
| author_facet | Jedlička, Přemysl Pilitowska, Agata |
| contents | We present a complete characterization of all indecomposable non-degenerate, not necessarily involutive, solutions of the Yang-Baxter equation of multipermutation level~2. We show that every such solution is a homomorphic image of a special, ``largest'' solution called \emph{the universal} one. On the other hand we prove that there is much simpler description. At first, on the product of a group $Z_n^2$ and an abelian group $G$, we construct some family of indecomposable non-degenerate solutions of the Yang-Baxter equation of multipermutation level $2$. Next, applying Rosenbaum's theorem of subgroups of a semidirect product and isolating a triple: a subgroup of $G$, a subgroup of $Z_n^2$ and one group homomorphism, we obtain a~full description of each epimorphism which gives the desired solutions. Such a construction provides a tool how to find (and possibly enumerate) all indecomposable non-degenerate solutions of multipermutation level $2$. We also argue that the automorphism group of the discussed solutions is regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation Jedlička, Přemysl Pilitowska, Agata Group Theory Primary: 16T25. Secondary: 20B35, 20E07 We present a complete characterization of all indecomposable non-degenerate, not necessarily involutive, solutions of the Yang-Baxter equation of multipermutation level~2. We show that every such solution is a homomorphic image of a special, ``largest'' solution called \emph{the universal} one. On the other hand we prove that there is much simpler description. At first, on the product of a group $Z_n^2$ and an abelian group $G$, we construct some family of indecomposable non-degenerate solutions of the Yang-Baxter equation of multipermutation level $2$. Next, applying Rosenbaum's theorem of subgroups of a semidirect product and isolating a triple: a subgroup of $G$, a subgroup of $Z_n^2$ and one group homomorphism, we obtain a~full description of each epimorphism which gives the desired solutions. Such a construction provides a tool how to find (and possibly enumerate) all indecomposable non-degenerate solutions of multipermutation level $2$. We also argue that the automorphism group of the discussed solutions is regular. |
| title | Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation |
| topic | Group Theory Primary: 16T25. Secondary: 20B35, 20E07 |
| url | https://arxiv.org/abs/2508.17981 |