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arXiv · 2204.01445

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

Kurusch Ebrahimi-Fard·Frédéric Patras·Nikolas Tapia·Lorenzo Zambotti
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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.

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Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia, Lorenzo Zambotti. 2022-04-04. Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability. https://doi.org/10.3842/sigma.2023.038

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