# Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

## TL;DR

This paper introduces a new algebraic structure on decorated planar rooted forests, defining a weighted infinitesimal unitary bialgebra and related cocycle conditions, extending the Connes-Kreimer Hopf algebra framework.

## Contribution

It constructs a free weighted infinitesimal unitary bialgebra on rooted forests and develops a new pre-Lie algebraic structure, generalizing existing algebraic frameworks.

## Key findings

- Defined a new coproduct on decorated forests.
- Proved the space of forests is the free weighted cocycle infinitesimal bialgebra.
- Constructed a new pre-Lie algebraic structure on forests.

## Abstract

In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of $\Omega$-cocycle infinitesimal bialgebras of weight $\lambda$ and then prove that the space of decorated planar rooted forests $H_{\mathrm{RT}}(X,\Omega)$, together with a set of grafting operations $\{ B^+_\omega \mid \omega\in \Omega\}$, is the free $\Omega$-cocycle infinitesimal unitary bialgebra of weight $\lambda$ on a set $X$, involving a weighted version of a Hochschild 1-cocycle condition. As an application, we equip a free cocycle infinitesimal unitary bialgebraic structure on the undecorated planar rooted forests, which is the object studied in the well-known (noncommutative) Connes-Kreimer Hopf algebra. Finally, we construct a new pre-Lie algebraic structure on decorated planar rooted forests.