Combinatorial Hopf algebras

TL;DR

This paper introduces the concept of combinatorial Hopf algebras, exploring their algebraic structures, examples like the Connes-Kreimer algebra, and their connections to operads such as dendriform and pre-Lie.

Contribution

It formalizes the notion of combinatorial Hopf algebras with specific isomorphisms and links them to operads, providing a unified framework for their study.

Findings

Characterization of cofree-cocommutative right-sided combinatorial Hopf algebras by primitive pre-Lie algebras

Identification of several operad triples related to these Hopf algebras

Analysis of the algebraic structures underlying key examples like the Connes-Kreimer algebra

Abstract

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.