The Baker-Campbell-Hausdorff formula via mould calculus

Abstract

The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{\'o} Ecalle's mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] 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.