  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.
