Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees

TL;DR

This paper extends the Connes-Moscovici Hopf algebra to include all rooted trees, bridging non-commutative geometry and quantum field theory, and enabling new cohomological studies.

Contribution

It introduces a generalized Hopf algebra that encompasses the entire rooted trees Hopf algebra, linking two important mathematical structures.

Findings

Established a relationship between the generalized algebra and rooted trees

Opened avenues for cohomology studies of rooted trees Hopf algebra

Bridged non-commutative geometry and quantum field theory

Abstract

This paper defines a generalization of the Connes-Moscovici Hopf algebra, $\mathcal{H}(1)$ that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the later, a much studied object in perturbative Quantum Field Theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.