Relating two Hopf algebras built from an operad

TL;DR

This paper explores the relationship between two Hopf algebras derived from an operad, establishing a surjective morphism between them and illustrating this with the example of rooted trees and the Connes-Kreimer algebra.

Contribution

It demonstrates a surjective morphism from the Hopf algebra of functions on a group constructed from an operad to the incidence Hopf algebra built from associated posets.

Findings

Existence of a surjective Hopf algebra morphism.

Identification of the incidence Hopf algebra with the Connes-Kreimer algebra in a specific case.

Connection between operad-based constructions and well-known Hopf algebras.

Abstract

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $\NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.