# The geometry of characters of Hopf algebras

## TL;DR

This paper explores the geometric and Lie group structures of character groups of Hopf algebras, introducing a novel Lie group structure for graded Hopf algebras and establishing their regularity.

## Contribution

It introduces a new Lie group structure for characters of graded Hopf algebras with finite-dimensional degree zero parts and proves their regularity.

## Key findings

- Established a Lie group structure for characters of graded Hopf algebras.
- Proved regularity of these Lie groups in Milnor's sense.
- Unified various series expansions within a geometric framework.

## Abstract

Character groups of Hopf algebras appear in a variety of mathematical contexts such as non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. In these applications, several species of "series expansions" can then be described as characters from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, Chen-Fliess series from control theory and rough paths. In this note we explain and review the constructions for Lie group and topological structures for character groups. The main novel result of the present article is a Lie group structure for characters of graded and not necessarily connected Hopf algebras (under the assumption that the degree zero subalgebra is finite-dimensional). Further, we establish regularity (in the sense of Milnor) for these Lie groups.