Finiteness conditions on skew braces and solutions of the Yang-Baxter equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914375131987968 |
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| author | Cascella, Rosa Properzi, Silvia Van Antwerpen, Arne |
| author_facet | Cascella, Rosa Properzi, Silvia Van Antwerpen, Arne |
| contents | A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $λ_f$-skew brace, i.e. every element has finitely many $λ$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $λ_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06177 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finiteness conditions on skew braces and solutions of the Yang-Baxter equation Cascella, Rosa Properzi, Silvia Van Antwerpen, Arne Group Theory Quantum Algebra 16T25, 81R50, 20F24, 20N99 A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $λ_f$-skew brace, i.e. every element has finitely many $λ$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $λ_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index. |
| title | Finiteness conditions on skew braces and solutions of the Yang-Baxter equation |
| topic | Group Theory Quantum Algebra 16T25, 81R50, 20F24, 20N99 |
| url | https://arxiv.org/abs/2603.06177 |