Right-handed Hopf algebras and the preLie forest formula

TL;DR

This paper extends the Zimmermann forest formula to right-handed polynomial Hopf algebras, which are dual to preLie algebras, simplifying antipode computations in combinatorial and quantum field theory contexts.

Contribution

It generalizes the forest formula to a broader class of Hopf algebras, specifically right-handed polynomial ones, linked to preLie algebra structures.

Findings

Forest formula applies to right-handed polynomial Hopf algebras.

The class of these Hopf algebras includes those of Feynman diagrams.

Simplifies antipode calculations in quantum field theory.

Abstract

Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it generalizes to arbitrary right-handed polynomial Hopf algebras. These Hopf algebras are dual to the enveloping algebras of preLie algebras -a structure common to many combinatorial Hopf algebras which is carried in particular by the Hopf algebras of Feynman diagrams.