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The Magnus Expansion, Trees and Knuth’s Rotation Correspondence

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Abstract

W. Magnus introduced a particular differential equation characterizing the logarithm of the solution of linear initial value problems for linear operators. The recursive solution of this differential equation leads to a peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli numbers, iterated Lie brackets and integrals. This paper aims at obtaining further insights into the fine structure of the Magnus expansion. By using basic combinatorics on planar rooted trees we prove a closed formula for the Magnus expansion in the context of free dendriform algebra. From this, by using a well-known dendriform algebra structure on the vector space generated by the disjoint union of the symmetric groups, we derive the Mielnik–Plebański–Strichartz formula for the continuous Baker–Campbell–Hausdorff series.

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Notes

  1. We recover the q=1 case of a q-analog formula by F. Chapoton (see [10, Proposition 5.10]).

References

  1. A. Agrachev, R. Gamkrelidze, Chronological algebras and nonstationary vector fields, J. Sov. Math. 17, 1650–1675 (1981).

    Article  Google Scholar 

  2. A.A. Agrachev, R.V. Gamkrelidze, The shuffle product and symmetric groups, in Differential Equations, Dynamical Systems and Control Science, ed. by K.D. Elworthy, W.N. Everitt, E.B. Lee. Lecture Notes in Pure and Appl. Math., vol. 152 (Dekker, New York, 1994), pp. 365–382.

    Google Scholar 

  3. F.V. Atkinson, Some aspects of Baxter’s functional equation, J. Math. Anal. Appl. 7, 1–30 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pac. J. Math. 10, 731–742 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Blanes, F. Casas, J.A. Oteo, J. Ros, Magnus expansion: mathematical study and physical applications, Phys. Rep. 470, 151–238 (2009).

    Article  MathSciNet  Google Scholar 

  6. Ch. Brouder, Â. Mestre, F. Patras, Tree expansion in time-dependent perturbation theory, J. Math. Phys. 51(7), 072104 (2010).

    Article  MathSciNet  Google Scholar 

  7. J.C. Butcher, An algebraic theory of integration methods, Math. Comput. 26, 79–106 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Cayley, On the theory of analytical forms called trees, Philos. Mag. 13, 172–176 (1857).

    Google Scholar 

  9. F. Chapoton, Rooted trees and an exponential-like series. arXiv:math/0209104.

  10. F. Chapoton, A rooted-trees q-series lifting a one-parameter family of Lie idempotents, Algebra Number Theory 3, 611–636 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Chapoton, F. Patras, Enveloping algebras of pre-Lie algebras, Solomon idempotents and the Magnus formula. arXiv:1201.2159v1 [math.QA].

  12. F. Chapoton, F. Hivert, J.C. Novelli, J.Y. Thibon, An operational calculus for the mould operad, Int. Math. Res. Not. 2008(9) (2008).

  13. F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001, 395–408 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  14. Ph. Chartier, E. Hairer, G. Vilmart, Algebraic structures of B-series, Found. Comput. Math. 10, 407–427 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  15. Ph. Chartier, A. Murua, An algebraic theory of order, Modél. Math. Anal. Numér. 43, 607–630 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198, 198–246 (1998).

    Article  MathSciNet  Google Scholar 

  17. G. Duchamp, F. Hivert, J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, J. Alg. Comput. 12, 671–717 (2002).

    MATH  MathSciNet  Google Scholar 

  18. K. Ebrahimi-Fard, D. Manchon, A magnus- and fer-type formula in dendriform algebras, Found. Comput. Math. 9, 295–316 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Ebrahimi-Fard, D. Manchon, Dendriform equations, J. Algebra 322, 4053–4079 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  20. K. Ebrahimi-Fard, D. Manchon, Twisted dendriform algebras and the pre-Lie magnus expansion, J. Pure Appl. Algebra 215, 2615–2627 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra 209(2), 439–459 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  22. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112, 218–348 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Gubinelli, Abstract integration, combinatorics of trees and differential equations, in Combinatorics and Physics. Contemp. Math., vol. 539 (AMS, Providence, 2011), pp. 135–151.

    Chapter  Google Scholar 

  24. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31 (Springer, Berlin, 2002).

    Book  Google Scholar 

  25. P. Hanlon, The fixed-point partition lattices, Pac. J. Math. 96(2), 319–341 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Hoffman, Combinatorics of rooted trees and Hopf algebras, Trans. Am. Math. Soc. 355, 3795–3811 (2003).

    Article  MATH  Google Scholar 

  27. A. Iserles, S.P. Nørsett, On the solution of linear differential equations in Lie groups, Philos. Trans. R. Soc. Lond. A 357, 983–1020 (1999).

    Article  MATH  Google Scholar 

  28. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie-group methods, Acta Numer. 9, 215–365 (2000).

    Article  Google Scholar 

  29. A. Iserles, Expansions that grow on trees, Not. Am. Math. Soc. 49, 430–440 (2002).

    MATH  MathSciNet  Google Scholar 

  30. N. Jacobson, Lie Algebras, 2nd edn. (Dover, New York, 1979).

    Google Scholar 

  31. D.E. Knuth, The Art of Computer Programming I. Fundamental Algorithms (Addison-Wesley, Reading, 1968).

    Google Scholar 

  32. J.-L. Loday, Dialgebras, Lect. Notes Math. 1763, 7–66 (2001).

    Article  MathSciNet  Google Scholar 

  33. J.-L. Loday, M. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139, 293–309 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  34. J.-L. Loday, M. Ronco, Order structure and the algebra of permutations and of planar binary trees, J. Algebr. Comb. 15(3), 253–270 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Lundervold, H. Munthe-Kaas, Hopf algebras of formal diffeomorphisms and numerical integration on manifolds, Contemp. Math. 539, 295–324 (2011).

    Article  MathSciNet  Google Scholar 

  36. A. Lundervold, H. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds. arXiv:1112.4465v1 [math.NA].

  37. T. Lyons, Differential equations driven by rough paths, Rev. Mat. Iberoam. 14, 215–310 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  38. W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649–673 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  39. C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177(3), 967–982 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Manchon, A short survey on pre-Lie algebras, in Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, ed. by A. Carey. E. Schrödinger Institut Lectures in Math. Phys. (Eur. Math. Soc., Zurich, 2011).

    Google Scholar 

  41. H. Munthe-Kaas, Lie–Butcher theory for Runge–Kutta methods, BIT Numer. Math. 35, 572–587 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  42. H. Munthe-Kaas, Runge–Kutta methods on Lie groups, BIT Numer. Math. 38, 92–111 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  43. B. Mielnik, J. Plebański, Combinatorial approach to Baker–Campbell–Hausdorff exponents, Ann. Inst. Henri Poincaré A XII, 215–254 (1970).

    Google Scholar 

  44. A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math. 6(4), 387–426 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  45. D. Segal, Free left-symmetric algebras and an analogue of the Poincaré–Birkhoff–Witt–theorem, J. Algebra 164, 750–772 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  46. F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Am. Math. Soc. 82, 323–339 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  47. R.S. Strichartz, The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations, J. Funct. Anal. 72, 320–345 (1987).

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We thank H. Munthe-Kaas and A. Lundervold for discussions and remarks. The first author is supported by a Ramón y Cajal research grant from the Spanish government. Both authors were supported by the CNRS (GDR Renormalisation).

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas, C/Nicolás Cabrera, no. 13-15, 28049, Madrid, Spain

    Kurusch Ebrahimi-Fard

  2. University Blaise Pascal, C.N.R.S.-UMR 6620, BP80026, 63171, Aubière Cedex, France

    Dominique Manchon

Authors
  1. Kurusch Ebrahimi-Fard
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  2. Dominique Manchon
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Correspondence to Dominique Manchon.

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Communicated by Andrew Odlizko.

K. Ebrahimi-Fard on leave from University de Haute Alsace, Mulhouse, France.

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Ebrahimi-Fard, K., Manchon, D. The Magnus Expansion, Trees and Knuth’s Rotation Correspondence. Found Comput Math 14, 1–25 (2014). https://doi.org/10.1007/s10208-013-9172-x

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  • DOI: https://doi.org/10.1007/s10208-013-9172-x

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