Mathematics

Post-Lie algebroids and Lie algebra actions

Ari Stern (Washington University in St. Louis)

Abstract: A fundamental problem in numerical analysis is the approximation of flows of vector fields on manifolds. For vector fields in Euclidean space (i.e., systems of ODEs), Butcher gave a brilliant characterization of Runge-Kutta methods in terms of the algebra and combinatorics of rooted trees. Subsequent work related this to the fact that vector fields on Euclidean space form a "pre-Lie algebra". More recently, this work has been extended to "post-Lie algebroids": vector bundles whose sections form a "post-Lie algebra" with respect to a connection. We give a complete characterization of post-Lie algebroids in terms of local Lie algebra actions. As a corollary, we get a "no-go theorem" limiting the cases to which this class of numerical methods can be applied. (Joint work with Hans Munthe-Kaas and Olivier Verdier.)