Two interacting Hopf algebras of trees

TL;DR

This paper introduces a new Hopf algebra structure on rooted forests and reveals a connection to the Connes-Kreimer algebra, with implications for numerical methods in differential equations.

Contribution

It constructs a novel Hopf algebra on rooted forests and links it to the Connes-Kreimer algebra via a bicomodule structure, unifying different algebraic approaches.

Findings

Established a Hopf algebra on rooted forests with a coproduct.

Linked the new algebra to the Connes-Kreimer algebra through a bicomodule structure.

Reinterpreted recent results in numerical methods for differential equations.

Abstract

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.