Coprime extensions of indecomposable solutions to the Yang-Baxter equation
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909571948216320 |
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| author | Dietzel, Carsten |
| author_facet | Dietzel, Carsten |
| contents | In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang--Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension. Furthermore, we define coprime extensions of solutions and prove that each coprime extension of indecomposable solutions can be obtained as a suitable twisted extension. We then apply our results to obtain a full description of indecomposable solutions of size $pqr$, where $p,q,r$ are different primes, from a structure theorem of Cedó and Okniński. We close with some remarks on a cohomology theory for solutions developed by Lebed and Vendramin. We express our results in the language of cycle sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coprime extensions of indecomposable solutions to the Yang-Baxter equation Dietzel, Carsten Quantum Algebra Group Theory Rings and Algebras 16T25, 20N02, 81R50 In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang--Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension. Furthermore, we define coprime extensions of solutions and prove that each coprime extension of indecomposable solutions can be obtained as a suitable twisted extension. We then apply our results to obtain a full description of indecomposable solutions of size $pqr$, where $p,q,r$ are different primes, from a structure theorem of Cedó and Okniński. We close with some remarks on a cohomology theory for solutions developed by Lebed and Vendramin. We express our results in the language of cycle sets. |
| title | Coprime extensions of indecomposable solutions to the Yang-Baxter equation |
| topic | Quantum Algebra Group Theory Rings and Algebras 16T25, 20N02, 81R50 |
| url | https://arxiv.org/abs/2411.11670 |