A survey on the Munthe-Kaas-Wright Hopf algebra (Q6632767)

The (commutative, non cocommutative) Munthe-Kaas-Wright Hopf algebra \(H_{MKW}\) of planar rooted trees is studied. Firstly, its dual is constructed, as the enveloping algebra of a free post-Lie algebra, using the Guin-Oudom construction. It is then given a natural growth operator, defined with graftings, similar to the one defined on the Butcher-Connes-Kreimer Hopf algebra. This new operation allows to prove the cofreeness of \(H_{MKW}\), and to obtain results on its endomorphisms and comodules. It is also proved that, as a consequence, \(H_{MKW}\) is isomorphic to a shuffle algebra over a certain alphabet. The consequences over planarly branched rough paths are explored, and in particular the fact that they can be expressed as geometric rough paths.