In this talk, we shall discuss an operadic perspective on K. Dykema’s twisted factorization formula for the operator-valued T-transform in free probability. To begin with, we introduce in the general setting of an operad with multiplication two group products (twisted Cauchy product and free product) on formal series of operators, besides the one introduced by F. Chapoton. We show that the twisted Cauchy product relates to the Cauchy product by means of a certain post-Lie structure, that exists on any operad with multiplication.
Furthermore, we explain how those products relate through a certain transformation which we call (abstract) T-transform borrowing terminology from free probability.
By specializing tp the endomorphism operad this construction gives a new perspective on the twisted factorization of Dykema's T-transform and to multiplicative free convolution.
We will discuss connections to the work of A. Frabetti and collaborators on an operadic approach to groups defined over Hopf algebras on rooted trees.