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

Semi-edges, reflections and Coxeter groups

Abstract. We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let π:Γ → (v, D, ι, −1) be a 1-covering of a monopole admitting semiedges only. The graph Γ is the Cayley graph of a Coxeter group if and only if π is regular and any deck transformation in ∆(π) that interchanges two neighboring vertices of Γ acts as a reflection on Γ. 1

Chemical mechanism development : laboratory studies and model application

Within the German Tropospheric Research Programme (TFS) numerous kinetic and mechanistic studies on the tropospheric reaction/degradation of the following reactants were carried out:oxygenated VOC,aromatic VOC,biogenic VOC,short-lived intermediates, such as alkoxy and alkylperoxy radicals.At the conception of the projects these selected groups were classes of VOC or intermediates for which the atmospheric oxidation mechanisms were either poorly characterised or totally unknown. The motivation for these studies was the attainment of significant improvements in our understanding of the atmospheric chemical oxidation processes of these compounds, particularly with respect to their involvement in photooxidant formation in the troposphere.In the present paper the types of experimental investigations performed and the results obtained within the various projects are briefly summarised. The major achievements are highlighted and discussed in terms of their contribution to improving our understanding of the chemical processes controlling photosmog formation in the troposphere