Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

TL;DR

This paper extends the preLie calculus for free, Boolean, and monotone moment-cumulant relations to two-faced non-commutative probability, enriching the mathematical framework for bi-free and related independences.

Contribution

It introduces a two-faced extension of preLie calculus for moment-cumulant relations, advancing the mathematical tools for bi-free and other two-faced independences.

Findings

Extended preLie calculus to two-faced non-commutative probability

Unified treatment of biBoolean, bifree, and bimonotone independences

Enhanced mathematical framework for multi-faced operator models

Abstract

One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is, computing distributions of operators in the algebra generated by the left- and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and type I bimonotone independences. In this paper, we extend the preLie calculus pertaining to free, Boolean, and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above-mentioned two-faced equivalents.