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

Abstract

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.

32 pages

Cited by in corpus (8)

• Relative Rota-Baxter groups and skew left braces
• What is the Magnus Expansion?
• Skew braces: a brief survey
• Cohomology and Extensions of Relative Rota-Baxter groups
• Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures
• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
• Formal integration of complete Rota-Baxter Lie algebras