Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

TL;DR

This paper unifies three distinct Hopf algebras from number theory, physics, and topology through simplicial and operadic frameworks, revealing common underlying structures and paving the way for new insights.

Contribution

It introduces a unified operadic and simplicial perspective connecting diverse Hopf algebras from different mathematical fields.

Findings

Unified framework for Hopf algebras from different areas

Identification of common operadic and simplicial structures

Potential for new constructions and reinterpretations

Abstract

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.