On Post-Lie Algebras, Lie–Butcher Series and Moving Frames

Abstract

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They 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 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.

Invariant Connections, Lie Algebra Actions, and Foundations of Numerical Integration on Manifolds

“…Recently, Munthe-Kaas and Lundervold [25] related this algebraic perspective to certain analytical techniques for approximating flows of vector fields: Butcher series methods (Butcher [5,6], Hairer and Wanner [15]) in the pre-Lie case and Lie-Butcher series methods (Munthe-Kaas [22,23,24]) in the more general post-Lie case. These techniques, which involve expressing flows as formal power series in rooted trees and forests, were originally developed for the analysis of numerical integrators.…”

“…To include these examples in the algebraic framework, Munthe-Kaas and Lundervold [25] considered connections more general than affine connections: namely, connections on a Lie algebroid A → M , which is an anchored vector bundle with a compatible Lie bracket on the space of sections Γ(A). (An affine connection is just the special case when A = T M is the tangent bundle with the Jacobi-Lie bracket.)…”

Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees

“…cuts γ Pc(t) X Rc(t) |ψ . [12] shows that planar rooted trees arise naturally in the case of flat connections with constant torsion.…”

“…t m−1 ), can be written as a linear combination of trees formed by repeated application of general natural growth operators. By (12), the tree t = B + (t 1 . .…”

Invariants of objects and their images under surjective maps

“…A wide range of image processing applications includes jigsaw puzzle assembly, [22], recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tumors, [19], recovering structure of three-dimensional objects from motion, [3], classification of pro-jective curves in visual recognition, [20], and construction of integral invariant signatures for object recognition in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, [5,27,28,40], symmetry and equivalence of polygons and point configurations, [8,25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34,35], the design and analysis of geometric integrators and symmetry-preserving numerical schemes, [26,37,47], the determination of Casimir invariants of Lie algebras and the classification of subalgebras, with applications in quantum mechanics, [7], and many more.…”