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The Baker–Campbell–Hausdorff formula via mould calculus

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Abstract

The well-known Baker–Campbell–Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product \(\text {e}^X \text {e}^Y\) can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction to Écalle’s mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin (Dokl Akad Nauk SSSR (NS) 57:323–326, 1947) explicit formula for the logarithm, as well as another formula recently obtained by Kimura (Theor Exp Phys 4:041A03, 2017) for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.

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Notes

  1. We assume \({\text {ord}}(A+B) \ge \min \{{\text {ord}}A,{\text {ord}}B\}\) and \({\text {ord}}(AB)\ge {\text {ord}}A + {\text {ord}}B\) for any \(A,B\in {\mathcal {A}}\), and \({\text {ord}}A = \infty \) iff \(A=0\).

  2. Indeed, \(\tau ^{-1}(i)\) is the position in \({\underline{n}}\) of \(\omega _i\), the ith letter of \({\underline{a}}\,{\underline{b}}\).

  3. In Écalle’s work, the initial motivation for the definition of alternality and symmetrality is the situation when \({\mathcal {A}}\) is an algebra of operators (acting on an auxiliary algebra) and each \(B_n\) acts as a derivation: in that case, the \(B_{[\,{\underline{n}}\,]}\)’s satisfy a modified Leibniz rule which involves the shuffling coefficients, whence it follows that MB is itself a derivation if M is an alternal mould, and an algebra automorphism if M is symmetral. Here we do not assume anything of that kind on \({\mathcal {A}}\) and the \(B_n\)’s but rather follow the spirit of “Lie mould calculus” as advocated in [20].

References

  1. Baumard, S., Schneps, L.: On the derivation representation of the fundamental Lie algebra of mixed elliptic motives. Annales Mathématiques du Québec 41(1), 43–62 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Behtash, A., Dunne, G.V., Schäfer, T., Sulejmanpasic, T., Ünsal, M.: Complexified path integrals, exact saddles, and supersymmetry. Phys. Rev. Lett. 116, 011601 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bonfiglioli, A., Fulci, R.: Topics in Noncommutative Algebra—The Theorem of Campbell, Baker, Hausdorff and Dynkin. Lecture Notes in Mathematics, vol. 2034. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  4. Bouillot, O., Écalle, J.: Invariants of identity-tangent diffeomorphisms expanded as series of multitangents and multizetas. In: Resurgence, Physics and Numbers, pp. 109–232, CRM Series, 20, Ed. Norm., Pisa (2017)

  5. Couso-Santamaría, R., Schiappa, R., Vaz, R.: On asymptotices and resurgent structures of enumerative Gromov–Witten invariants. Commun. Numer. Theor. Phys. 11, 707–790 (2017). arXiv:1605.07473

    Article  MATH  Google Scholar 

  6. Couso-Santamaría, R., Mariño, M., Schiappa, R.: Resurgence matches quantization. J. Phys. A Math. Theor. 50, 145402 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Dynkin, E.B.: Calculation of the coefficients in the Campbell-Hausdorff formula (Russian). Dokl. Akad. Nauk SSSR (N.S.) 57, 323–326 (1947)

    Google Scholar 

  8. Dillinger, H., Delabaere, E., Pham, F.: Résurgence de Voros et périodes des courbes hyperelliptiques. Annales de l’institut Fourier 43(1), 163–199 (1993)

    Article  MATH  Google Scholar 

  9. Écalle, J.: Les fonctions résurgentes, Publ. Math. d’Orsay [vol. 1: 81-05, vol. 2: 81-06, vol. 3: 85-05] (1981, 1985)

  10. Écalle, J.: Cinq applications des fonctions résurgentes. Publ. Math. d’Orsay, pp. 84–62 (1984)

  11. Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actualités Math, Hermann (1992)

    MATH  Google Scholar 

  12. Écalle, J.: ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan. Journal de Théorie des Nombres de Bordeaux 15(2), 411–478 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fauvet, F., Foissy, L., Manchon, D.: The Hopf algebra of finite topologies and mould composition. Annales de l’Institut Fourier 67(3), 911–945 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kimura, T.: Explicit description of the Zassenhaus formula. Theor. Exp. Phys. 4, 041A03 (2017)

    Google Scholar 

  16. Kontsevich, M.: Resurgence and Quantization. Course given at IHES, Paris in April 2017

  17. Matone, M.: An algorithm for the Baker–Campbell–Hausdorff formula. J. High Energy Phys. 05, 113 (2015). arXiv:1502.06589

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Menous, F.: Formal differential equations and renormalization. In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois theories, IRMA Lect. Math. Theor. Phys., vol. 15, pp. 229–246 (2009)

  19. Novelli, J.-C., Paul, T., Sauzin, D., Thibon, J.-Y.: Rayleigh-Schrödinger series and Birkhoff decomposition. Lett. Math. Phys. 108, 18 (2018). https://doi.org/10.1007/s11005-017-1040-1

    Article  MATH  Google Scholar 

  20. Paul, T., Sauzin, D.: Normalization in Lie algebras via mould calculus and applications. Regul. Chaotic Dyn. 22(6), 616–649 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Reutenauer, C.: Free Lie Algebras, London Mathematical Society Monographs, vol. 7. Clarendon Press, New York (1993)

    Google Scholar 

  22. Sauzin, D.: Initiation to mould calculus through the example of saddle-node singularities. Rev. Semin. Iberoam. Mat. 3(5–6), 147–160 (2008)

    MathSciNet  Google Scholar 

  23. Sauzin, D.: Mould expansions for the saddle-node and resurgence monomials. In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois Theories. IRMA Lectures in Mathematics and Theoretical Physics, vol. 15, pp. 83–163. European Mathematical Society, Zürich (2009)

    Chapter  Google Scholar 

  24. Sauzin, D.: Introduction to 1-summability and resurgence. In: Mitschi, C., Sauzin, D. (eds.) Divergent Series, Summability and Resurgence. I. Monodromy and Resurgence. Lecture Notes in Mathematics, vol. 2153. Springer, Berlin (2016)

    Google Scholar 

  25. Schneps, L.: Double shuffle and Kashiwara–Vergne Lie algebras. J. Algebra 367, 54–74 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thibon, J.-Y.: Noncommutative symmetric functions and combinatorial Hopf algebras. In: Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation, vol. I, pp. 219–258. CRM Series, 12, Ed. Norm., Pisa (2011)

  27. Voros, A.: The return of the quartic oscillator. The complex WKB method. Annales de l’I. H. P. Section A tome 39(3), 211–338 (1983)

    MathSciNet  MATH  Google Scholar 

  28. von Waldenfels, W.: Zur Charakterisierung Liescher Elemente in freien Algebren. Arch. Math. (Basel) 17, 44–48 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

D.S. and Y.L. thank the Centro Di Ricerca Matematica Ennio De Giorgi and the Scuola Normale Superiore di Pisa for their kind hospitality, during which this work was completed.

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China

    Yong Li & Shanzhong Sun

  2. CNRS UMR 8028 IMCCE, 77 av. Denfert-Rochereau, 75014, Paris, France

    David Sauzin

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  1. Yong Li
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  2. David Sauzin
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  3. Shanzhong Sun
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Correspondence to David Sauzin.

Additional information

Yong Li and Shanzhong Sun: Partially supported by NSFC (Nos. 11131004, 11271269, 11771303).

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Li, Y., Sauzin, D. & Sun, S. The Baker–Campbell–Hausdorff formula via mould calculus. Lett Math Phys 109, 725–746 (2019). https://doi.org/10.1007/s11005-018-1125-5

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