Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
Probability
2023-06-09 v3 Operator Algebras
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.
Keywords
Cite
@article{arxiv.2204.01445,
title = {Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability},
author = {Kurusch Ebrahimi-Fard and Frédéric Patras and Nikolas Tapia and Lorenzo Zambotti},
journal= {arXiv preprint arXiv:2204.01445},
year = {2023}
}