A Survey on the Munthe-Kaas-Wright Hopf Algebra

We survey the Munthe-Kaas--Wright Hopf algebra defined on planar rooted trees. This algebra serves a role akin to that of the Butcher--Connes--Kreimer Hopf algebra on non-planar rooted trees within the domain of numerical methods for ordinary differential equations. In the course of our presentation, we revisit Foissy's work on finite-dimensional comodules over the Butcher--Connes--Kreimer Hopf algebra and expand on his findings to include the Munthe-Kaas--Wright Hopf algebra. This involves detailing its endomorphisms and recursively constructing its primitive elements. These results are applied within the context of rough paths, where we describe an isomorphism between planarly branched and geometric rough paths. Our approach hinges on the extension of the Guin--Oudom construction to post-Lie algebras. In the case of the free post-Lie algebra defined on planar rooted trees, it yields the dual of the Munthe-Kaas--Wright Hopf algebra. Surprisingly, we uncover a natural connection between the concept of bialgebras in cointeraction and the Guin--Oudom construction. Prompted by the rough path perspective, we explore this finding through the lens of translations on rough paths. Additionally, we investigate the geometric embedding for planar regularity structures.