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

TL;DR

This paper reviews recent advances in understanding the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups, highlighting their applications in quantum field theory and numerical analysis.

Contribution

It provides a comprehensive overview of the conditions under which Hopf algebra character groups form (pro-)Lie groups and explores their relevance in various mathematical and physical contexts.

Findings

Character groups can form infinite-dimensional (pro-)Lie groups under mild conditions.

These structures have significant applications in quantum field theory renormalization.

They also play a role in numerical analysis, exemplified by the Butcher group.

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.