Overview of (pro-)Lie group structures on hopf algebra character groups
Paper i proceeding, 2018

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: • the Butcher group from numerical analysis and • character groups which arise from the Connes–Kreimer theory of renormalisation of quantum field theories.

Infinite-dimensional Lie group

Hopf algebra

pro-Lie group

Regular Lie group

Weakly complete space

Locally convex algebra

Butcher group


Geir Bogfjellmo

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Rafael Dahmen

Technische Universität Darmstadt

Alexander Schmeding

Norges teknisk-naturvitenskapelige universitet

Springer Proceedings in Mathematics and Statistics

2194-1009 (ISSN) 2194-1017 (eISSN)

Vol. 267 287-314


Algebra och logik


Matematisk analys



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