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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 |
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Main Authors: | , , , |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
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Summary: | 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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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.2204.01445 |