The Geometry of Characters of Hopf Algebras

  • Geir Bogfjellmo
  • Alexander SchmedingEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coalgebra that is also equipped with an antiautomorphism known as the antipode, satisfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra morphisms from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups.



This research was partially supported by the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691070 and by the Knut and Alice Wallenberg Foundation grant agreement KAW 2014.0354. We are indebted to K.-H. Neeb and R. Dahmen for discussions which led to Lemma 10. Further, we would like to thank L. Zambotti and Y. Bruned for explaining their results on character groups in the renormalisation of SPDEs. Finally, we thank K.H. Hofmann for encouraging and useful comments and apologize to him for leaving out [28] at first.


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Authors and Affiliations

  1. 1.Matematiska vetenskaperChalmers tekniska högskola och Göteborgs universitetGöteborgSweden
  2. 2.Institutt for matematiske fagNTNU TrondheimTrondheimNorway

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