Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 006, 50 pages      arXiv:2201.11747
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Joscha Diehl a, Malte Gerhold ab and Nicolas Gilliers ac
a) Universität Greifswald, Institut für Mathematik und Informatik, Walther-Rathenau-Str. 47, 17489 Greifswald, Germany
b) Norwegian University of Science and Technology (NTNU), Department of Mathematical Sciences, 7491 Trondheim, Norway
c) Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse, France

Received March 30, 2022, in final form January 10, 2023; Published online January 31, 2023

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.

Key words: shuffle algebras; non-commutative probability; cumulants; multi-faced; Möbius category.

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