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

• Published in: Foundations of computational mathematics Vol. 13; no. 4; pp. 583 - 613
• Main Authors: Munthe-Kaas, Hans Z., Lundervold, Alexander
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2013; Springer; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Computation; Computational mathematics; Computer Science; Economics; Frames; Invariants; Joints; Lie algebras; Lie groups; Linear and Multilinear Algebras; Manifolds; Math Applications in Computer Science; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Sequences (Mathematics); Series; Topological manifolds; Norway; United States; Connections; Post-Lie algebras; Lie group integrators; B-series; 53C; 65L; 16T; Lie–Butcher series; Homogeneous spaces; Pre-Lie algebras; Rooted trees; Post-Lie algebroids; Moving frames; Combinatorial Hopf algebras
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9167-7

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