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Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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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, combinatorial Hopf algebra, pre-Lie algebra.
Mots-clés : moments and cumulants 
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Kurusch Ebrahimi-Fard; Frédéric Patras; Nikolas Tapia; Lorenzo Zambotti. Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a37/

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