Apparaît dans la collection : Wall-Crossing Structures, Analyticity, and Resurgence

I will review the definition of the algebra A of simple Z-resurgent series and its alien derivations $\Delta_m$, as given by Jean Ecalle in 1981. In particular, I will recall why one can say that the alien derivations are independent in a strong sense. Then I will explore one consequence of the freeness of the Lie algebra generated by the $\Delta_m$'s under commutators and multiplication by elements of A: since we have so many derivations (although we are dealing with a formal series of _one_ variable), one can construct non-trivial Poisson structures on A and, correspondingly, non-commutative deformations of the product of A.