Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915711896518656 |
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| author | Albano, Andrea Stefanelli, Paola |
| author_facet | Albano, Andrea Stefanelli, Paola |
| contents | We introduce a novel algebraic structure called di-skew brace by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang-Baxter equation. We study the structural properties of these solutions with a particular focus on their left derived shelves, which belong to the class of conjugation racks. Consistently, we show that these solutions belong to a broader class that includes skew brace solutions. In particular, we prove that each such solution can be decomposed as a hemi-semidirect product of a skew brace solution endowed with a certain compatible action on the idempotents of the associated di-skew brace structure. Finally, we provide concrete instances of these solutions through a suitable notion of averaging operators on groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation Albano, Andrea Stefanelli, Paola Quantum Algebra 16T25, 81R50, 16Y99, 20N99, 20M99, 17B38 We introduce a novel algebraic structure called di-skew brace by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang-Baxter equation. We study the structural properties of these solutions with a particular focus on their left derived shelves, which belong to the class of conjugation racks. Consistently, we show that these solutions belong to a broader class that includes skew brace solutions. In particular, we prove that each such solution can be decomposed as a hemi-semidirect product of a skew brace solution endowed with a certain compatible action on the idempotents of the associated di-skew brace structure. Finally, we provide concrete instances of these solutions through a suitable notion of averaging operators on groups. |
| title | Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation |
| topic | Quantum Algebra 16T25, 81R50, 16Y99, 20N99, 20M99, 17B38 |
| url | https://arxiv.org/abs/2505.15387 |