Mixtures of classical and free independence

TL;DR

This paper revisits Lambda-freeness, a concept describing a mixture of classical and free independence in algebraic structures, providing a cumulant-based description to facilitate future quantum symmetry analysis.

Contribution

It offers a cumulant-based characterization of Lambda-freeness, bridging classical and free independence concepts for algebraic structures.

Findings

Lambda-freeness characterized via cumulants

Foundation for analyzing quantum symmetries in mixed independence

Revives and formalizes a classical-free independence concept

Abstract

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered.