Numerical integration methods using Lie group actions

Many of the manifolds used in applications are homogeneous spaces and are thus naturally characterized in terms of Lie group actions. In recent years, it has become increasingly popular to design integration methods for ODE's on such manifolds by including the group action as a part of the numerical scheme. The study of these methods involves various parts of the classical Lie algebra theory. One can divide the integration methods into two classes, those which work in the ``free setting'' and whose format and implementation are essentially independent of the particular Lie algebra used in the formulation. In this case the interesting object to study is a certain free Lie algebra, and its universal enveloping algebra which can be furnished with a Hopf algebra structure. The second class of methods are designed to make particular use of the structure of the associated Lie algebra, and their derivation make extensive use of root spaces and structure theory. In this talk we will survey modern techniques for analysing numerical integration methods based on Lie group actions by means of classical Lie algebra theory.