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

TL;DR

This paper introduces post-Lie algebras, generalizing pre-Lie algebras, and demonstrates their foundational role in Lie--Butcher series for analyzing flows on manifolds, with applications in differential geometry and numerical analysis.

Contribution

It defines post-Lie algebras, explores their relation to Lie--Butcher series, and develops new computational formulas based on recursions in a magma.

Findings

Post-Lie algebras generalize pre-Lie algebras for homogeneous spaces.

Lie--Butcher series are founded on post-Lie algebras.

New recursive formulas for free post-Lie and D-algebras are developed.

Abstract

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively 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 are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the 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. Lie--Butcher series combine Butcher series with Lie series and 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, are 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.