# On the pre-Lie algebra of specified Feynman graphs

## TL;DR

This paper explores the algebraic structures of specified Feynman graphs, introducing pre-Lie structures and their relations, and studies the associated enveloping algebras and module-bialgebra properties.

## Contribution

It defines a pre-Lie structure on the doubling space of specified Feynman graphs and establishes relations between different pre-Lie algebras and their enveloping algebras.

## Key findings

- velops a pre-Lie structure on the doubling space of graphs
- Proves the module property of the doubling space over the graph algebra
- Shows the enveloping algebra of one pre-Lie algebra forms a module-bialgebra

## Abstract

We study the pre-Lie algebra of specified Feynman graphs $\wt{V}_{\Cal T}$ and we define a pre-Lie structure on its doubling space $\wt{\Cal F}_{\Cal T}$. We prove that $\wt{\Cal F}_{\Cal T}$ is pre-Lie module on $\wt{V}_{\Cal T}$ and we find some relations between the two pre-Lie structures. Also, we study the enveloping algebras of two pre-Lie algebras denoted respectively by $(\wt {\Cal D'}_{\Cal T}, \bigstar, \Phi)$ and $(\wt {\Cal H'}_{\Cal T}, \star, \Psi)$ and we prove that $(\wt {\Cal D'}_{\Cal T}, \bigstar, \Phi)$ is a module-bialgebra on $(\wt {\Cal H'}_{\Cal T}, \star, \Psi)$.