Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914724179869696 |
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| author | Bai, Chengming Guo, Li Sheng, Yunhe Tang, Rong |
| author_facet | Bai, Chengming Guo, Li Sheng, Yunhe Tang, Rong |
| contents | The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang-Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the $\mathcal{P}$-group of an operad $\mathcal{P}$ naturally admit a pre-group structure. Next a relative Rota-Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota-Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang-Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie-Butcher group from numerical integration on manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation Bai, Chengming Guo, Li Sheng, Yunhe Tang, Rong Quantum Algebra Rings and Algebras 22E60, 16T25, 17B38, 65L99, 16T05, 18M60 The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang-Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the $\mathcal{P}$-group of an operad $\mathcal{P}$ naturally admit a pre-group structure. Next a relative Rota-Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota-Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang-Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie-Butcher group from numerical integration on manifolds. |
| title | Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation |
| topic | Quantum Algebra Rings and Algebras 22E60, 16T25, 17B38, 65L99, 16T05, 18M60 |
| url | https://arxiv.org/abs/2306.11196 |