On the Hopf Algebraic Structure of Lie Group Integrators

A commutative but not cocommutative graded Hopf algebra HN , based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees HC , developed by Butcher in his study of Runge–Kutta methods and later rediscovered by Connes and Moscovici in the context of non-commutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that HN is naturally obtained from a universal ob ject in a category of non- commutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on non-commutative Lie group actions on a manifold. Recursive and non-recursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of HN is a Hopf al- gebra of Grossman and Larson. HN contains two well-known Hopf algebras as special cases: The Hopf algebra HC of Butcher–Connes–Kreimer is identi- fied as a proper subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebra HF of the Free Associative Algebra is obtained from HN by a quotient construction.