Doubling pre-Lie algebra of rooted trees

TL;DR

This paper explores the structure of pre-Lie algebras associated with rooted trees, introduces a new pre-Lie structure on their doubling space, and investigates their enveloping algebras and interrelations.

Contribution

It defines a pre-Lie structure on the doubling space of rooted trees and analyzes the enveloping algebras, revealing their module-bialgebra relationship.

Findings

The doubling space $(V, ightarrow)$ admits a new pre-Lie structure.

The enveloping algebras $( ext{ extcal{H}}', igstar, ext{ extcal{G}})$ and $( ext{ extcal{D}}', igstar, ext{ extcal{ extchi}})$ are explicitly constructed.

$( ext{ extcal{D}}', igstar, ext{ extcal{ extchi}})$ is a module-bialgebra over $( ext{ extcal{H}}', igstar, ext{ extcal{ extG}})$.

Abstract

We study the pre-Lie algebra of rooted trees $(\Cal T, \rightarrow)$ and we define a pre-Lie structure on its doubling space $(V, \leadsto)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted respectively by $(\Cal H', \star, \Gamma)$ and $(\Cal D', \bigstar, \chi)$. We prove that $(\Cal D', \bigstar, \chi)$ is a module-bialgebra on $(\Cal H', \star, \Gamma)$ and we find some relations between the two pre-Lie structures.