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Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation

Bai, Chengming ; Guo, Li ; Sheng, Yunhe ; Tang, Rong

Mathematische annalen, 2024-01, Vol.388 (3), p.3127-3167 [Periódico revisado por pares]

Berlin/Heidelberg: Springer Berlin Heidelberg

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  • Título:
    Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation
  • Autor: Bai, Chengming ; Guo, Li ; Sheng, Yunhe ; Tang, Rong
  • Assuntos: Algebra ; Braiding ; Fields (mathematics) ; Lie groups ; Mathematics ; Mathematics and Statistics ; Numerical integration ; Operators (mathematics) ; Vector spaces ; Vectors (mathematics)
  • É parte de: Mathematische annalen, 2024-01, Vol.388 (3), p.3127-3167
  • Descrição: 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 -group of an operad 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.
  • Editor: Berlin/Heidelberg: Springer Berlin Heidelberg
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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