Recently, the interest in the structure of cumulants and Wick products for non-Gaussian variables has been revived, since they both play important roles in M. Hairer's theory of regularity structures. We present a new approach $ (group actions, Hopf algebras) that allow to understand the theory of Wick products as a deformation, a presentation from which many of their properties follow easily. We also discuss how these ideas appear in the theory of regularity structu$ Tapia, L. Zambotti.