Letters in Mathematical Physics

, Volume 104, Issue 10, pp 1281–1302 | Cite as

The Pre-Lie Structure of the Time-Ordered Exponential

  • Kurusch Ebrahimi-FardEmail author
  • Frédéric Patras


The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work, we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota–Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust–Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials.

Mathematics Subject Classification (2010)

05E15 16T05 16T30 


pre-Lie algebra time-ordering time-ordered exponential Rota–Baxter algebra Grossman–Larson algebra Bohnenblust–Spitzer identity Baker–Campbell–Hausdorff formula 


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  1. 1.
    Agrachev A., Gamkrelidze R.: Chronological algebras and nonstationary vector fields. J. Sov. Math. 17, 1650–1675 (1981)CrossRefGoogle Scholar
  2. 2.
    Agrachev, A., Gamkrelidze, R.: The shuffle product and symmetric groups. In: Elworthy, K.D., et al. (eds.) Differential Equations, Dynamical Systems, and Control Science. A Festschrift in Honor of Lawrence Markus. Lecture Notes in Pure and Applied Mathematics, vol. 152. Marcel Dekker, New York, pp. 365–382 (1994)Google Scholar
  3. 3.
    Aguiar M., Moreira W.: Combinatorics of the free Baxter algebra. Electron. J. Combin. 13(1), R17 (2006)MathSciNetGoogle Scholar
  4. 4.
    Baxter G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731–742 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brouder Ch., Patras F.: Hyperoctahedral Chen calculus for effective Hamiltonians. J. Algebra 322, 4105–4120 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cartier P.: On the structure of free Baxter algebras. Adv. Math. 9, 253–265 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cartier, P.: Vinberg algebras, Lie groups and combinatorics. In: Clay Mathematics Proceedings. Quanta of Maths, vol. 11, pp. 107–126. American Mathematical Society, Providence (2010)Google Scholar
  8. 8.
    Chapoton F.: Un théorème de Cartier–Milnor–Moore–Quillen pour les bigèbres dendriformes et les algèbres braces. J. Pure Appl. Algebra 168, 1–18 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chapoton F.: A rooted-trees q-series lifting a one-parameter family of Lie idempotents. Algebra Number Theory 3, 611–636 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chapoton F., Livernet M.: Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 8, 395–408 (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Chapoton, F., Livernet, M.: Relating two Hopf algebras built from an operad. Int. Math. Res. Not. 24, 27 pp (2007)Google Scholar
  12. 12.
    Chapoton F., Patras F.: Enveloping algebras of pre-Lie algebras, Solomon idempotents and the Magnus formula. Int. J. Algebra Comput. 23, 853–861 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Connes A., Kreimer D.: Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ebrahimi-Fard K., Guo L.: Free Rota–Baxter algebras and rooted trees. J. Algebra Appl. 7, 167–194 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81, 61–75 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ebrahimi-Fard K., Manchon D., Patras F.: A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion. J. Noncommut. Geom. 3, 181–222 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ebrahimi-Fard K., Manchon D.: A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. 9, 295–316 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ebrahimi-Fard K., Manchon D.: The Magnus expansion, trees and Knuth’s rotation correspondence. Found. Comput. Math. 14, 1–25 (2014)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Ebrahimi-Fard K., Patras F.: La structure combinatoire du calcul intégral [The combinatorial structure of integral calculus]. Gaz. Math. No. 138, 5–22 (2013)MathSciNetGoogle Scholar
  20. 20.
    Eilenberg S., Mac Lane S.: Cohomology theory of abelian groups and homotopy theory III. Proc. Natl. Acad. Sci. USA 37, 307–310 (1951)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Eilenberg S., Mac Lane S.: On the Groups H(π, n) I. Ann. Math. (2) 58, 55–106 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Foissy L.: Bidendriform bialgebras, trees, and free quasi-symmetric functions. J. Pure Appl. Algebra 209, 439–459 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Foissy L.: Free brace algebras are free pre-Lie algebras. Commun. Algebra 38, 3358–3369 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Gelfand I.M., Krob D., Lascoux A., Leclerc B., Retakh V., Thibon J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Grossman R., Larson R.: Hopf algebraic structures of families of trees. J. Algebra 26, 184–210 (1989)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Guo L., Keigher W.: Baxter algebras and shuffle products. Adv. Math. 150, 117–149 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2002)Google Scholar
  28. 28.
    Hoffman M.E.: Quasi-Shuffle Products. J. Algebr. Comb. 11, 49–68 (2000)CrossRefzbMATHGoogle Scholar
  29. 29.
    Lada T.: L algebra morphisms and symmetric brace algebras. J. Math. Sci. 186, 766–769 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Lam C.S.: Decomposition of time-ordered products and path-ordered exponentials. J. Math. Phys. 39, 5543–5558 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Manchon, D.: A short survey on pre-Lie algebras. In: Carey, A. (ed.) Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory. E. Schrödinger Institute Lectures in Mathematics and Physics. European Mathematical Society (2011)Google Scholar
  33. 33.
    Mielnik B., Plebański J.: Combinatorial approach to Baker–Campbell–Hausdorff exponents. Ann. Inst. H. Poincaré A (N.S.) 12, 215–254 (1970)zbMATHGoogle Scholar
  34. 34.
    Novelli J.-C., Thibon J.-Y.: A one-parameter family of dendriform identities. J. Combin. Theory Ser. A 116, 864–874 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Oudom J.-M., Guin D.: On the Lie enveloping algebra of a pre-Lie algebra. J. K-Theory 2, 147–167 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Patras F.: L’algèbre des descentes d’une bigèbre graduée. J. Algebra 170, 547–566 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Reutenauer C.: Free Lie Algebras. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  38. 38.
    Rota G.-C.: Baxter algebras and combinatorial identities. I. Bull. Am. Math. Soc. 75, 325–329 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Rota G.-C.: Baxter algebras and combinatorial identities. II. Bull. Am. Math. Soc. 75, 330–334 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Rota, G.-C., Smith, D.: Fluctuation theory and Baxter algebras. In: Symposia Mathematica, vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome 1971), pp. 179–201. Academic Press, London (1972)Google Scholar
  41. 41.
    Rota G.-C.: Baxter operators, an introduction. In: Kung, J.P.S. (ed.) Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries. Contemporary Mathematicians. Birkhäuser Boston, Boston (1995)Google Scholar
  42. 42.
    Rota, G.-C.: Ten mathematics problems I will never solve, Invited address at the joint meeting of the American Mathematical Society and the Mexican Mathematical Society, Oaxaca, Mexico, 6 Dec. 1997. Mitt. Dtsch. Math.-Ver. 2, 45–52 (1998)Google Scholar
  43. 43.
    Strichartz R.S.: The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations. J. Funct. Anal. 72, 320–345 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.ICMATMadridSpain
  2. 2.Univ. de Nice, Labo. J.-A. Dieudonné, UMR 7351, CNRSNice Cedex 02France

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