Overview of (pro-)Lie group structures on Hopf algebra character groups
arXiv:1711.05963 · doi:10.1007/978-3-030-01397-4_8
Abstract
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.
arXiv comment31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spain
References in corpus (9)
- Hopf Algebras, Renormalization and Noncommutative Geometry
- A Lie theoretic approach to renormalization
- Renormalization of gauge fields: A Hopf algebra approach
- On post-Lie algebras, Lie--Butcher series and moving frames
- An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
- Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators
- Differentiable mappings on products with different degrees of differentiability in the two factors
- The structure of renormalization Hopf algebras for gauge theories I: Representing Feynman graphs on BV-algebras
- The Lie group structure of the Butcher group