Foundations of Computational Mathematics

, Volume 13, Issue 4, pp 583–613 | Cite as

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

  • Hans Z. Munthe-Kaas
  • Alexander Lundervold


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.


B-series Combinatorial Hopf algebras Connections Homogeneous spaces Lie group integrators Lie–Butcher series Moving frames Post-Lie algebras Post-Lie algebroids Pre-Lie algebras Rooted trees 

Mathematics Subject Classification

65L 53C 16T 



We would like to thank Kurusch Ebrahimi-Fard, Dominique Manchon and Olivier Verdier for valuable discussions regarding the topics of this paper, and thanks to Jon Eivind Vatne for explaining operads and for pointing us to the work of Bruno Vallette. Finally, many thanks to Matthias Kawski and the anonymous referees for their careful reading, and for their many suggested improvements of the original manuscript.


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Copyright information

© SFoCM 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Inria Bordeaux Sud-OuestBordeauxFrance

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