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

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A Correction to this article was published on 24 September 2018

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Abstract

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.

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Change history

  • 24 September 2018

    The correct formula for the dimension of graded components

Notes

  1. The MC form can also be defined by right translation, but the left form is more convenient for moving frames.

  2. Trees with different orderings of the branches are considered different, as when pictured in the plane.

  3. Various notations for similar grafting products are found in the literature, e.g. u▷v=u[v]=u↷v.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Z. Munthe-Kaas

  2. Inria Bordeaux Sud-Ouest, Bordeaux, France

    Alexander Lundervold

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  1. Hans Z. Munthe-Kaas
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  2. Alexander Lundervold
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Correspondence to Hans Z. Munthe-Kaas.

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Communicated by Arieh Iserles.

Dedicated to Peter Olver in celebration of his 60th birthday.

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Munthe-Kaas, H.Z., Lundervold, A. On Post-Lie Algebras, Lie–Butcher Series and Moving Frames. Found Comput Math 13, 583–613 (2013). https://doi.org/10.1007/s10208-013-9167-7

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