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This article is cited in 1 scientific paper (total in 1 paper)
Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
Kurusch Ebrahimi-Farda, Frédéric Patrasb, Nikolas Tapiacd, Lorenzo Zambottie a Department of Mathematical Sciences, Norwegian University of Science and Technology,
NO 7491 Trondheim, Norway
b Université Côte d'Azur, CNRS, UMR 7351, Parc Valrose, 06108 Nice Cedex 02, France
c Weierstraß-Institut Berlin, Berlin, Germany
d Technische Universität Berlin, Berlin, Germany
e LPSM, Sorbonne Université, CNRS, Université Paris Cité, 4 Place Jussieu, 75005 Paris, France
Abstract:
We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.
Keywords:
non-commutative probability theory, non-commutative power series, moments and cumulants, combinatorial Hopf algebra, pre-Lie algebra.
Received: April 5, 2022; in final form May 29, 2023; Published online June 8, 2023
Citation:
Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia, Lorenzo Zambotti, “Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability”, SIGMA, 19 (2023), 038, 17 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1933 https://www.mathnet.ru/eng/sigma/v19/p38
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