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Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability (2204.01445v3)
Published 4 Apr 2022 in math.PR and math.OA
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.