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

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.