Letters in Mathematical Physics

, Volume 109, Issue 3, pp 725–746 | Cite as

The Baker–Campbell–Hausdorff formula via mould calculus

  • Yong Li
  • David SauzinEmail author
  • Shanzhong Sun


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.


Baker-Campbell-Hausdorff theorem Dynkin formula Mould calculus 

Mathematics Subject Classification

22E15 17B05 17B60 17B81 



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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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsCapital Normal UniversityBeijingPeople’s Republic of China
  2. 2.CNRS UMR 8028 IMCCEParisFrance

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