Schröder trees and antipode formulas: An application to non-commutative probability and Wick polynomials

Abstract

The double tensor Hopf algebra has been introduced by Ebrahimi-Fard and Patras to provide an algebraic framework for cumulants in non-commutative probability theory. In this paper, we obtain a cancellation-free formula, represented in terms of Schröder trees, for the antipode in the double tensor Hopf algebra. We apply the antipode formula to recover cumulant-moment formulas as well as a new expression for Anshelevich’s free Wick polynomials in terms of Schröder trees.