Diagonals of solutions of the Yang-Baxter equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910643045531648 |
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| author | Jedlicka, Premysl Pilitowska, Agata |
| author_facet | Jedlicka, Premysl Pilitowska, Agata |
| contents | We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: ``Is every non-degenerate solution bijective?'' of Cedó, Jespers and Verwimp. Additionally, we show that for a subclass of solutions called k-permutational, only one-sided non-degeneracy suffices to prove that one of the diagonal mappings is invertible. We also present an equational characterization of multipermutation solutions and extend results of Rump and Gateva-Ivanova about decomposability to non-involutive case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diagonals of solutions of the Yang-Baxter equation Jedlicka, Premysl Pilitowska, Agata Rings and Algebras Primary: 16T25. Secondary: 08A05, 20B30 We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: ``Is every non-degenerate solution bijective?'' of Cedó, Jespers and Verwimp. Additionally, we show that for a subclass of solutions called k-permutational, only one-sided non-degeneracy suffices to prove that one of the diagonal mappings is invertible. We also present an equational characterization of multipermutation solutions and extend results of Rump and Gateva-Ivanova about decomposability to non-involutive case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable. |
| title | Diagonals of solutions of the Yang-Baxter equation |
| topic | Rings and Algebras Primary: 16T25. Secondary: 08A05, 20B30 |
| url | https://arxiv.org/abs/2402.15652 |