Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)


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 Locally convex algebra Butcher group Weakly complete space pro-Lie group Regular Lie group 

MSC 2010

22E65 (primary) 16T05 43A40 58B25 46H30 22A05 (Secondary) 



The research on this paper was partially supported by the project Topology in Norway (NRC project 213458) and Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics (NRC project 231632). We thank J. M. Sanz-Serna for pointing out references to results from numerical analysis which the authors were unaware of. Finally, we thank the anonymous referees for many useful comments which helped to improve the manuscript.


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Authors and Affiliations

  1. 1.Chalmers Technical University & Gothenburg UniversityGothenburgSweden
  2. 2.TU DarmstadtDarmstadtGermany
  3. 3.Department of Mathematical SciencesNorwegian University of Science and Technology—NTNUTrondheimNorway

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