Combinatorial Hopf algebras from renormalization

  title={Combinatorial Hopf algebras from renormalization},
  author={Christian Brouder and Alessandra Frabetti and Fr{\'e}d{\'e}ric Menous},
  journal={Journal of Algebraic Combinatorics},
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field.We also describe two general ways to define the… 

Combinatorial Hopf algebras

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra

Non-perturbative graph languages, halting problem and complexity

Abstract We explain the foundations of a new class of formal languages for the construction of large Feynman diagrams which contribute to solutions of all combinatorial Dyson–Schwinger equations in a

Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras

The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions

Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization

The present work contains a consistent formulation of the methods of dimensional regularization (DimReg) and minimal subtraction (MS) in Minkowski position space. The methods are implemented into the

A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory

This monograph aims to build some new mathematical structures originated from Dyson--Schwinger equations for the description of non-perturbative aspects of gauge field theories whenever bare or

Graphons and renormalization of large Feynman diagrams

The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions.

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of

Weighted infinitesimal bialgebras

As a uniform of two versions of infinitesimal bialgebras introduced respectively by Joni-Rota and Loday-Ronco, weighted infinitesimal bialgebras play an important role in mathematics and mathematical



Combinatorial Hopf algebras

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra

On the structure of cofree Hopf algebras

Abstract We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a

Operads and the Hopf algebras of renormalisation

Functors from (co)operads to bialgebras relate Hopf algebras that occur in renormalisation to operads, which simplifies the proof of the Hopf algebra axioms, and induces a characterisation of the

QED Hopf algebras on planar binary trees

On the Lie envelopping algebra of a pre-Lie algebra

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We

Renormalization as a functor on bialgebras

On the Hopf algebra structure of perturbative quantum field theories

We show that the process of renormalization encapsules a Hopf algebra structure in a natural manner. This sheds light on the recently proposed connection between knots and renormalization theory.

Hopf-algebraic structure of families of trees