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Wick polynomials in noncommutative probability: a group-theoretical approach

Published online by Cambridge University Press:  25 August 2021

Kurusch Ebrahimi-Fard
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway e-mail:
Frédéric Patras
Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNR, UMR 7351, Parc Valrose, Nice, France e-mail:
Nikolas Tapia*
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany and Institute of Mathematics, Technische Universtät Berlin, Straße des 17. Juni 136, 10587 Berlin, Germany
Lorenzo Zambotti
Laboratoire de Probabilités, Statistiques et Modélisation, Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France e-mail:


Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

© Canadian Mathematical Society 2021

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This work was supported by the European Research Council for Informatics and Mathematics through contract ERCIM 2018-10, and the BMS MATH+ EF1-5 project “On robustness of Deep Neural Networks.”


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