On post-Lie algebras, Lie-Butcher series and moving frames

Details On post-Lie algebras, Lie-Butcher series and moving frames On post-Lie algebras, Lie-Butcher series and moving frames

2012-03-21

arxiv.org/abs/1203.4738v1

Mathematics

Numerical Analysis

20 pages

Scientific paper

Pre-Lie (or Vinberg) algebras arise from a flat and torsion free connection on a differential manifold. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series is an algebraic tool used to study geometric properties of flows on euclidean spaces, which is founded on pre-Lie algebras. Motivated by analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. The generalized Lie-Butcher series combining Butcher series with Lie series are used to analyze flows on manifolds. In this paper we show that Lie-Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, is explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie-Butcher series are related to invariants of curves described by moving frames.

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