A Generalization of Connes-Kreimer Hopf Algebra

TL;DR

This paper introduces 'bonsai' Hopf algebras as a generalization of Connes-Kreimer algebras, exploring their operad structures, new differentials, and cohomologies related to Feynman diagrams and renormalization.

Contribution

It presents a novel class of Hopf algebras called bonsais, establishes their operad structure, and investigates their differentials and cohomologies, extending the framework of Connes-Kreimer algebras.

Findings

Bonsai Hopf algebras generalize Connes-Kreimer algebras.

Operad structures are established on bonsais.

A new differential inspired by tree structures is introduced and analyzed.

Abstract

``Bonsai'' Hopf algebras, introduced here, are generalizations of Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization. We show that we can find operad structure on the set of bonsais. We introduce a new differential on these bonsai Hopf algebras, which is inspired by the tree differential. The cohomologies of these are computed here, and the relationship of this differential with the appending operation $*$ of Connes-Kreimer Hopf algebras is investigated.