The geometry of characters of hopf algebras
Paper in proceeding, 2018

Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coalgebra that is also equipped with an antiautomorphism known as the antipode, satisfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra morphisms from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups.

Author

Geir Bogfjellmo

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Alexander Schmeding

Norwegian University of Science and Technology (NTNU)

Abel Symposia

2193-2808 (ISSN) 2197-8549 (eISSN)

Vol. 13 159-185

The Abel Symposium, 2016
Rosendal, Norway,

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1007/978-3-030-01593-0_6

More information

Latest update

3/4/2019 9