3 research outputs found
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
Manifolds of mappings for continuum mechanics
This is an overview article. After an introduction to convenient calculus in
infinite dimensions, the foundational material for manifolds of mappings is
presented. The central character is the smooth convenient manifold
of all smooth mappings from a finite dimensional Whitney
manifold germ into a smooth manifold . A Whitney manifold germ is a
smooth (in the interior) manifold with a very general boundary, but still
admitting a continuous Whitney extension operator. This notion is developed
here for the needs of geometric continuum mechanics.Comment: 66 pages. arXiv admin note: substantial text overlap with
arXiv:1505.02359, arXiv:math/9202208. Last version: some misprintscorrecte