Algebraic Structures on Ordered Rooted Trees and Their Significance to Lie Group Integrators

Most Lie group integrators can be expanded in series indexed by the set of ordered rooted trees. To each tree one can associate two distinct higher order derivation operators, which we call frozen and unfrozen operators. Composition of frozen operators induces a concatenation product on the trees, whereas composition of unfrozen operators induces a somewhat more complicated product known as the Grossman--Larson product. Both of these algebra structures can be supplemented by the same coalgebra structure and an antipode, the result being two distinct cocommutative graded Hopf algebras. We discuss the use of these structures and characterize subsets of the Hopf algebras corresponding to vector fields and mappings on manifolds. This is further relevant for deriving order conditions for a general class of Lie group integrators and for deriving the modified vector field in backward error analysis for these integrators.