Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
| Autor: | Bai, Chengming, Guo, Li, Sheng, Yunhe, Tang, Rong |
|---|---|
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Math. Ann. 388 (3) (2024), 3127-3167 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00208-023-02592-z |
| Popis: | 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. Comment: 32 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|