Overview of (pro-)Lie group structures on Hopf algebra character groups

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.Comment: 31 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, Spai

Taylor approximations of operator functions

This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.Comment: 12 page

Multiplicative Renormalization and Hopf Algebras

Contains fulltext : 84013.pdf (preprint version ) (Open Access

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra

Rationality of spectral action for Robertson-Walker metrics

We use parametric pseudodifferential calculus to prove a conjecture by A. Connes and A. Chamseddine: we show that each term in the heat kernel expansion of the Dirac-Laplacian of a Robertson-Walker metric is described a several variable polynomial with rational coefficients