(Non)Commutative Hopf algebras of trees and (quasi)symmetric functions

TL;DR

This paper explores the relationships between various Hopf algebras of trees and symmetric functions, providing a framework that simplifies computations in combinatorial quantum field theory.

Contribution

It establishes commutative diagrams linking these Hopf algebras, offering new insights and computational tools for Dyson-Schwinger equations.

Findings

Unified diagrammatic framework for Hopf algebras of trees and symmetric functions

Simplified computations in the Connes-Kreimer Hopf algebra

Enhanced understanding of algebraic structures in quantum field theory

Abstract

The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.