Communications in Number Theory and Physics

Volume 14 (2020)

Number 1

From Möbius inversion to renormalisation

Pages: 171 – 198



Joachim Kock (Departement de Mathemàtiques, Universitat Autònoma de Barcelona, Spain)


This paper traces a straight line from classical Möbius inversion to Hopf-algebraic perturbative renormalisation. This line, which is logical but not entirely historical, consists of just a few main abstraction steps, and some intermediate steps dwelled upon for mathematical pleasure. The paper is largely expository, but contains many new perspectives on well-known results. For example, the equivalence between the Bogoliubov recursion and the Atkinson formula is exhibited as a direct generalisation of the equivalence between the Weisner–Rota recursion and the Hall–Leroux formula for Möbius inversion.


Möbius inversion, perturbative renormalisation, bialgebras, coalgebras

2010 Mathematics Subject Classification

Primary 11A25, 81T15. Secondary 16T10, 16T15.

Support from grants MTM2016-80439-P (AEI/FEDER, UE) of Spain and 2017-SGR-1725 of Catalonia is gratefully acknowledged.

Received 19 September 2018

Accepted 27 August 2019

Published 22 January 2020