Hopf algebras of rooted forests, cocyles and free Rota-Baxter algebras

TL;DR

This paper explores the algebraic structures of rooted forests, cocycles, and free Rota-Baxter algebras, revealing universal properties and their applications to quantum field theory renormalization.

Contribution

It introduces the concept of cocycle Hopf algebras and applies universal properties to structure free Rota-Baxter algebras as cocycle Hopf algebras.

Findings

Universal properties of decorated rooted trees with 1-cocycles

Construction of cocycle Hopf algebras from rooted forests

Application to free Rota-Baxter algebras

Abstract

The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra or a cocycle bialgebra.