On the Hopf Algebraic Structure of Lie Group Integrators

Abstract

A commutative but not cocommutative graded Hopf algebra HN, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of 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 noncommutative 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 object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between HN and four other Hopf algebras is discussed. The dual of HN is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra HC of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra HSh is obtained from HN by a quotient construction. The Hopf algebra HP of ordered trees by Foissy differs from HN in the definition of the product (noncommutative concatenation for HP and shuffle for HN) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.