In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker-Campbell-Hausdorff-Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group.If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge-Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker-Campbell-Hausdorff Lie group modelled on a Fréchet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated to the Butcher group by Connes and Kreimer.

In this article, a unified approach to obtain symplectic integrators on T * G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge-Kutta-Munthe-Kaas methods and Crouch-Grossman methods. These methods can be interpreted as symplectic partitioned Runge-Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.

Aromatic B-series are a generalization of B-series. Some of the operations defined for B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.

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.