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

TL;DR

This paper explores a new algebraic structure on non-commutative power series, linking group laws and pre-Lie structures to deepen understanding of identities in free probability theory.

Contribution

It introduces a novel group law on formal power series in non-commuting variables and connects it to pre-Lie structures, providing a new framework for free probability identities.

Findings

Established a group law induced by linear forms on a Hopf algebra.

Recast key free probability identities in a group-theoretic framework.

Connected non-commutative power series structures to pre-Lie algebra concepts.

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.