Braided dynamical groups, the dynamical Yang-Baxter equation and related structures
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arXiv
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| Format: | Preprint |
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2025
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| author | Bai, Chengming Guo, Li Sheng, Yunhe Wang, You |
| author_facet | Bai, Chengming Guo, Li Sheng, Yunhe Wang, You |
| contents | We introduce the notion of a braided dynamical group which is a matched pair of dynamical groups satisfying extra conditions. It is shown to give a solution of the dynamical Yang-Baxter equation and at the same time a braided groupoid, thereby integrating the approaches of Andruskiewitsch and Matsumoto-Shimizu respectively that use these two notions to produce quiver-theoretical solutions of the Yang-Baxter equation. We pursue this connection further by relative Rota-Baxter operators on dynamical groups, which give rise to matched pairs of dynamical groups. As the derived structures of relative Rota-Baxter operators on dynamical groups, dynamical post-groups are introduced and are shown to be equivalent to braided dynamical groups. Finally, skew-braces are generalized to dynamical skew-braces as another equivalent notion of braided dynamical groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Braided dynamical groups, the dynamical Yang-Baxter equation and related structures Bai, Chengming Guo, Li Sheng, Yunhe Wang, You Mathematical Physics Group Theory Rings and Algebras We introduce the notion of a braided dynamical group which is a matched pair of dynamical groups satisfying extra conditions. It is shown to give a solution of the dynamical Yang-Baxter equation and at the same time a braided groupoid, thereby integrating the approaches of Andruskiewitsch and Matsumoto-Shimizu respectively that use these two notions to produce quiver-theoretical solutions of the Yang-Baxter equation. We pursue this connection further by relative Rota-Baxter operators on dynamical groups, which give rise to matched pairs of dynamical groups. As the derived structures of relative Rota-Baxter operators on dynamical groups, dynamical post-groups are introduced and are shown to be equivalent to braided dynamical groups. Finally, skew-braces are generalized to dynamical skew-braces as another equivalent notion of braided dynamical groups. |
| title | Braided dynamical groups, the dynamical Yang-Baxter equation and related structures |
| topic | Mathematical Physics Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2509.21708 |