## 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

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

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

Various notations for similar grafting products are found in the literature, e.g.

*u*▷*v*=*u*[*v*]=*u*↷*v*.

## References

A.A. Agrachev, R.V. Gamkrelidze, Chronological algebras and nonstationary vector fields,

*J. Math. Sci.***17**(1), 1650–1675 (1981).H. Berland, Isotropy in geometric integration. PhD thesis, Master’s thesis, Norwegian University of Science and Technology (2002).

H. Berland, B. Owren, Algebraic structures on ordered rooted trees and their significance to Lie group integrators, in

*Group Theory and Numerical Analysis*. CRM Proceedings & Lecture Notes, vol. 39 (AMS, Providence, 2005), pp. 49–63.C. Brouder, Runge–Kutta methods and renormalization,

*Eur. Phys. J. C***12**(3), 521–534 (2000).J.C. Butcher, Coefficients for the study of Runge–Kutta integration processes,

*J. Aust. Math. Soc.***3**(02), 185–201 (1963).J.C. Butcher, An algebraic theory of integration methods,

*Math. Comput.***26**(117), 79–106 (1972).É.E. Cartan, J.A. Schouten, On the geometry of the group-manifold of simple and semi-simple groups. Koninklijke Akademie van Wetenschappen te Amsterdam (1926).

A. Cayley, On the theory of the analytical forms called trees,

*Philos. Mag. Ser. 4***13**(85) (1857).E. Celledoni, B. Owren, On the implementation of Lie group methods on the Stiefel manifold,

*Numer. Algorithms***32**(2), 163–183 (2003).F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad,

*Int. Math. Res. Not.***2001**(8), 395–408 (2001).A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry,

*Commun. Math. Phys.***199**(1), 203–242 (1998).P.E. Crouch, R. Grossman, Numerical integration of ordinary differential equations on manifolds,

*J. Nonlinear Sci.***3**(1), 1–33 (1993).M. Degeratu, M. Ivan, Linear connections on lie algebroids, in

*Proceedings of the 5th Conference of Balkan Society of Geometers*(BSG, Bucharest, 2006), pp. 44–53.K. Ebrahimi-Fard, D. Manchon, The Magnus expansion, trees and Knuth’s rotation correspondence. arXiv:1203.2878 (2012).

M. Fels, P.J. Olver, Moving coframes: I. A practical algorithm,

*Acta Appl. Math.***51**(2), 161–213 (1998).M. Fels, P.J. Olver, Moving coframes: II. Regularization and theoretical foundations,

*Acta Appl. Math.***55**(2), 127–208 (1999).R.B. Gardner,

*The Method of Equivalence and Its Applications*(SIAM, Philadelphia, 1989).M. Gerstenhaber, The cohomology structure of an associative ring,

*Ann. Math.***78**(2), 267–288 (1963).M. Goze, E. Remm, Lie-admissible algebras and operads,

*J. Algebra***273**(1), 129–152 (2004).R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees,

*J. Algebra***126**(1), 184–210 (1989).E. Hairer, G. Wanner, On the Butcher group and general multi-value methods,

*Computing***13**(1), 1–15 (1974).A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie-group methods,

*Acta Numer.***9**, 215–365 (2000).N. Jacobson,

*Lie Algebras*(Dover, New York, 1979).S. Kobayashi, K. Nomizu,

*Foundations of Differential Geometry*, vol. 2 (Interscience, New York, 1969).D. Lewis, P.J. Olver, Geometric integration algorithms on homogeneous manifolds,

*Found. Comput. Math.***2**(4), 363–392 (2002).D. Lewis, N. Nigam, P.J. Olver, Connections for general group actions,

*Commun. Contemp. Math.***7**, 341–374 (2005).J.L. Loday, M.O. Ronco, Combinatorial Hopf algebras, in

*Quanta of Maths*. Clay Mathematics Proceedings, vol. 11 (2010).O. Loos,

*Symmetric Spaces: General Theory*, vol. 1 (WA Benjamin, 1969).A. Lundervold, H.Z. Munthe-Kaas, Backward error analysis and the substitution law for Lie group integrators,

*Found. Comput. Math.*1–26 (2011).A. Lundervold, H.Z. Munthe-Kaas, Hopf algebras of formal diffeomorphisms and numerical integration on manifolds,

*Contemp. Math.***539**, 295–324 (2011).A. Lundervold, H.Z. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds, in

*IRMA Lectures in Mathematics and Theoretical Physics*(2013).K.C.H. Mackenzie,

*General Theory of Lie Groupoids and Lie Algebroids*, vol. 213 (Cambridge University Press, Cambridge, 2005).E.L. Mansfield,

*A Practical Guide to the Invariant Calculus*(Cambridge University Press, Cambridge, 2010).H. Munthe-Kaas, Lie–Butcher theory for Runge–Kutta methods,

*BIT Numer. Math.***35**(4), 572–587 (1995).H. Munthe-Kaas, Runge–Kutta methods on Lie groups,

*BIT Numer. Math.***38**(1), 92–111 (1998).H. Munthe-Kaas, High order Runge–Kutta methods on manifolds,

*Appl. Numer. Math.***29**(1), 115–127 (1999).H. Munthe-Kaas, S. Krogstad, On enumeration problems in Lie–Butcher theory,

*Future Gener. Comput. Syst.***19**(7), 1197–1205 (2003).H. Munthe-Kaas, B. Owren, Computations in a free Lie algebra,

*Philos. Trans. R. Soc., Math. Phys. Eng. Sci.***357**(1754), 957 (1999).H. Munthe-Kaas, W. Wright, On the Hopf algebraic structure of Lie group integrators,

*Found. Comput. Math.***8**(2), 227–257 (2008).H. Munthe-Kaas, A. Zanna, Numerical integration of differential equations on homogeneous manifolds, in

*Foundations of Computational Mathematics*, ed. by F. Cucker, M. Shub (1997).J.C. Novelli, J.Y. Thibon, Parking functions and descent algebras,

*Ann. Comb.***11**(1), 59–68 (2007).P.J. Olver,

*Equivalence, Invariants, and Symmetry*(Cambridge University Press, Cambridge, 1995).P.J. Olver, A survey of moving frames,

*Comput. Algebra Geom. Algebra Appl.*105–138 (2005).B. Owren, A. Marthinsen, Runge–Kutta methods adapted to manifolds and based on rigid frames,

*BIT Numer. Math.***39**(1), 116–142 (1999).C. Reutenauer,

*Free Lie Algebras*(Oxford University Press, London, 1993).R.W. Sharpe,

*Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program*(Springer, Berlin, 1997).N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences. http://oeis.org/A022553.

M. Spivak,

*A Comprehensive Introduction to Differential Geometry*, vol. 2, 3rd edn. Publish or Perish (2005).B. Vallette, Homology of generalized partition posets,

*J. Pure Appl. Algebra***208**(2), 699–725 (2007).E.B. Vinberg, The theory of convex homogeneous cones,

*Trans. Mosc. Math. Soc.***12**, 340–403 (1963).A. Zanna, H.Z. Munthe-Kaas, Generalized polar decompositions for the approximation of the matrix exponential,

*SIAM J. Matrix Anal. Appl.***23**(3), 840–862 (2002).

## 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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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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DOI: https://doi.org/10.1007/s10208-013-9167-7

### Keywords

- 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