Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

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...

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Published in:arXiv.org 2023-06
Main Authors: Ebrahimi-Fard, Kurusch, Patras, Frédéric, Tapia, Nikolas, Zambotti, Lorenzo
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Language:English
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Group theory
Mathematical analysis
Power series
Probability theory
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Patras, Frédéric
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description 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.
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subjects Group theory
Mathematical analysis
Power series
Probability theory
title Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
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