Skew braces and Rota-Baxter operators on semi-direct products
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| Format: | Preprint |
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2025
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| _version_ | 1866918419880738816 |
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| author | Belwal, Pragya Singh, Mahender |
| author_facet | Belwal, Pragya Singh, Mahender |
| contents | This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace, referred to as its square, on the natural semi-direct product of its additive and multiplicative groups. Further, the square construction is distinct from the previously known double construction arising as a special case of matched pairs of skew braces. This provides a method to construct a new bijective, non-degenerate solution to the Yang--Baxter equation from an existing solution arising from a skew left brace. We show that the square construction is functorial and integrates naturally into both the cohomological and extension-theoretic frameworks for (relative) Rota--Baxter groups and skew left braces. Furthermore, we provide a sufficient condition under which two isoclinic skew left braces yield isoclinic squares. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13959 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew braces and Rota-Baxter operators on semi-direct products Belwal, Pragya Singh, Mahender Quantum Algebra Group Theory Rings and Algebras 17B38, 16T25, 81R50 This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace, referred to as its square, on the natural semi-direct product of its additive and multiplicative groups. Further, the square construction is distinct from the previously known double construction arising as a special case of matched pairs of skew braces. This provides a method to construct a new bijective, non-degenerate solution to the Yang--Baxter equation from an existing solution arising from a skew left brace. We show that the square construction is functorial and integrates naturally into both the cohomological and extension-theoretic frameworks for (relative) Rota--Baxter groups and skew left braces. Furthermore, we provide a sufficient condition under which two isoclinic skew left braces yield isoclinic squares. |
| title | Skew braces and Rota-Baxter operators on semi-direct products |
| topic | Quantum Algebra Group Theory Rings and Algebras 17B38, 16T25, 81R50 |
| url | https://arxiv.org/abs/2509.13959 |