Lie–Butcher Series, Geometry, Algebra and Computation

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)


Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell.


B-series Lie–Butcher series Post-Lie algebra Pre-Lie algebra 


16T05 17B99 17D99 65D30 


  1. 1.
    Benettin, G., Giorgilli, A.: On the hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74(5–6), 1117–1143 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Butcher, J.C.: Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc. 3(02), 185–201 (1963)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26(117), 79–106 (1972)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Calaque, D., Ebrahimi-Fard, K., Manchon, D.: Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series. Adv. Appl. Math. 47(2), 282–308 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cayley, A.: On the theory of the analytical forms called trees. Philos. Mag. 13(19), 4–9 (1857)Google Scholar
  6. 6.
    Chapoton, F., Livernet, M.: Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 2001(8), 395–408 (2001)Google Scholar
  7. 7.
    Chartier, P., Hairer, E., Vilmart, G.: Numerical integrators based on modified differential equations. Math. Comput. 76(260), 1941 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chartier, P., Hairer, E., Vilmart, G.: Algebraic structures of B-series. Found. Comput. Math. 10(4), 407–427 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dzhumadil’daev, A., Löfwall, C.: Trees, free right-symmetric algebras, free Novikov algebras and identities. Homol. Homotopy Appl. 4(2), 165–190 (2002)Google Scholar
  10. 10.
    Ebrahimi-Fard, K., Gracia-Bondía, J.M., Patras, F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276(2), 519–549 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ebrahimi-Fard, K., Lundervold, A., Mencattini, I., Munthe-Kaas, H.Z.: Post-Lie algebras and isospectral flows. Symmetry Integr. Geom. Methods Appl. (SIGMA) 11(93) (2015)Google Scholar
  12. 12.
    Ebrahimi-Fard, K., Lundervold, A., Munthe-Kaas, H.: On the Lie enveloping algebra of a post-Lie algebra. J. Lie Theory 25(4), 1139–1165 (2015)Google Scholar
  13. 13.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31. Springer series in computational mathematics (2006)Google Scholar
  14. 14.
    Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13(1), 1–15 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numer. 2000(9), 215–365 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lundervold, A., Munthe-Kaas, H.: Backward error analysis and the substitution law for Lie group integrators. Found. Comput. Math. 13(2), 161–186 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Munthe-Kaas, H.: Lie-Butcher theory for Runge-Kutta methods. BIT Numer. Math. 35(4), 572–587 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Munthe-Kaas, H.: Runge-Kutta methods on Lie groups. BIT Numer. Math. 38(1), 92–111 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Munthe-Kaas, H., Krogstad, S.: On enumeration problems in Lie-Butcher theory. Future Gen. Comput. Syst. 19(7), 1197–1205 (2003)CrossRefGoogle Scholar
  20. 20.
    Munthe-Kaas, H., Owren, B.: Computations in a free Lie algebra. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 357(1754), 957–981 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Munthe-Kaas, H.Z., Lundervold, A.: On post-Lie algebras, Lie-butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Munthe-Kaas, H.Z., Wright, W.M.: On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8(2), 227–257 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Oudom, J.-M., Guin, D.: On the Lie enveloping algebra of a pre-Lie algebra. J. K-Theory K-Theory Appl. Algebra Geom. Topol. 2(01), 147–167 (2008)Google Scholar
  24. 24.
    Owren, B., Marthinsen, A.: Runge-Kutta methods adapted to manifolds and based on rigid frames. BIT Numer. Math. 39(1), 116–142 (1999)Google Scholar
  25. 25.
    Philippe C., Hairer, E., Vilmart, G.: A substitution law for B-series vector fields. Technical Report 5498, INRIA (2005)Google Scholar
  26. 26.
    Reutenauer, C.: Free Lie Algebras. Oxford University Press (1993)Google Scholar
  27. 27.
    Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

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