Communications in Number Theory and Physics

Volume 14 (2020)

Number 1

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

Pages: 1 – 90



Imma Gálvez-Carrillo (Departament de Matemàtiques, Universitat Politècnica de Catalunya, Terrassa, Spain)

Ralph M. Kaufmann (Department of Mathematics and Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana, U.S.A.)

Andrew Tonks (Department of Mathematics, University of Leicester, United Kingdom)


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.

R.K. gratefully acknowledges support from the Humboldt Foundation and the Simons Foundation, the Institut des Hautes Etudes Scientifiques and the Max–Planck–Institut for Mathematics in Bonn and the University of Barcelona for their support.

I.G.C. was partially supported by Spanish Ministry of Science and Catalan government grants MTM2012-38122-C03-01, MTM2013-42178-P, 2014- SGR-634, MTM2015-69135-P, MTM2016-76453-C2-2-P (AEI/FEDER, UE), MTM2017-90897-REDT, and 2017-SGR-932

A.T. was supported by grants MTM2013-42178-P, and MTM2016-76453-C2-2-P (AEI/FEDER, UE).

Received 18 June 2018

Accepted 27 August 2019

Published 22 January 2020