A group-theoretical approach to conditionally free cumulants

TL;DR

This paper extends a group-theoretical framework to conditionally free cumulants in non-commutative probability, revealing new algebraic structures and relations that deepen understanding of these cumulants.

Contribution

It introduces a non-cocommutative Hopf algebra approach to conditionally free cumulants, expanding the algebraic tools for non-commutative probability theory.

Findings

Development of a non-cocommutative Hopf algebra for cumulants

Extension of classical group-Lie algebra relations

Identification of new adjoint actions in the algebraic structure

Abstract

In this work we extend the recently introduced group-theoretical approach to moment-cumulant relations in non-commutative probability theory to the notion of conditionally free cumulants. This approach is based on a particular combinatorial Hopf algebra which may be characterised as a non-cocommutative generalisation of the classical unshuffle Hopf algebra. Central to our work is the resulting non-commutative shuffle algebra structure on the graded dual. It implies an extension of the classical relation between the group of Hopf algebra characters and its Lie algebra of infinitesimal characters and, among others, the appearance of new forms of adjoint actions of the group on its Lie algebra which happens to play a key role in the new algebraic understanding of conditionally free cumulants.