On the pre-Lie algebra of specified Feynman graphs

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 $(\wt {\Cal H'}_{\Cal T}, \star, \Psi)$.