Infinitesimal unitary Hopf algebras and planar rooted forests

TL;DR

This paper explores infinitesimal unitary Hopf algebras, establishing their structure on decorated planar rooted forests and demonstrating their universality as free cocycle infinitesimal unitary Hopf algebras.

Contribution

It introduces a new approach to infinitesimal unitary Hopf algebras using Hochschild cocycle conditions and characterizes planar rooted forests as free objects in this category.

Findings

Decorated planar rooted forests form the free cocycle infinitesimal unitary bialgebra.

Planar rooted forests are the free cocycle infinitesimal unitary Hopf algebra on the empty set.

The approach contrasts with Aguiar's by focusing on infinitesimal unitary structures.

Abstract

Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal bialgebra is at the same time an algebra and coalgebra, in such a way that the comultiplication is a derivation. Twenty years after Joni and Rota, Aguiar introduced the concept of an infinitesimal (non-unitary) Hopf algebra. In this paper we study infinitesimal unitary bialgebras and infinitesimal unitary Hopf algebras, in contrary to Aguiar's approach. Using an infinitesimal version of the Hochschild 1-cocycle condition, we prove respectively that a class of decorated planar rooted forests is the free cocycle infinitesimal unitary bialgebra and free cocycle infinitesimal unitary Hopf algebra on a set. As an application, we obtain that the planar rooted forests is the free cocycle infinitesimal unitary Hopf algebra on the empty set.