Mixtures of classical and free independence

We revive the concept of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda}$$\end{document}-freeness of Młotkowski (Quant Prob Relat Top 7:27–41, 2004), 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 (Speicher and Weber, Quantum groups with partial commutation relations, 2016) where the quantum symmetries underlying these mixtures of classical and free independences will be considered.