A nondiagrammatic description of the Connes-Kreimer Hopf algebra

TL;DR

This paper reveals that the core algebraic structure of the Connes-Kreimer Hopf algebra aligns with the canonical pre-Lie structure of polynomial vector fields, leading to a tensor-based Hopf algebra equivalent to the original.

Contribution

It provides a nondiagrammatic, algebraic perspective connecting the Connes-Kreimer Hopf algebra to polynomial vector fields, offering a new construction via tensors.

Findings

Identifies the insertion pre-Lie structure with polynomial vector fields

Constructs a tensor-based Hopf algebra isomorphic to the Connes-Kreimer algebra

Bridges algebraic structures with quantum field theory renormalization

Abstract

We demonstrate that the fundamental algebraic structure underlying the Connes-Kreimer Hopf algebra -- the insertion pre-Lie structure on graphs -- corresponds directly to the canonical pre-Lie structure of polynomial vector fields. Using this fact, we construct a Hopf algebra built from tensors that is isomorphic to a version of the Connes-Kreimer Hopf algebra that first appeared in the perturbative renormalization of quantum field theories.