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

The notions of a post-group and pre-group are introduced as unification enrichment several group structures appearing in diverse areas from numerical integration to the Yang–Baxter equation. First Butcher on Euclidean spaces -group an operad naturally admit structure. Next relative Rota–Baxter operator splits structure Conversely, gives rise sub-adjacent group. Further braided solution Indeed category post-groups is isomorphic groups skew-left braces. Moreover post-Lie differentiates algebra vector space left invariant fields, showing that integral objects algebras. Finally, post-Hopf algebras Magnus expansions utilized study formal As byproduct, explicitly determined Lie–Butcher manifolds.