On the pre-Lie algebra of specified Feynman graphs

Abstract

We study the pre-Lie algebra of specified Feynman graphs ṼT and we define a pre-Lie structure on its doubling space F̃T. We prove that F̃T is pre-Lie module on ṼT 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 (D′̃T,★,Φ) and (H′̃T,⋆,Ψ) and we prove that (D′̃T,★,Φ) is a module-bialgebra on (H′̃T,⋆,Ψ).