Homogeneous Lie Groups and Quantum Probability

TL;DR

This paper generalizes the algebro-geometric approach to free probability, connecting convolution products, moment-cumulant relations, and Lie group structures within non-commutative probability theories.

Contribution

It extends the geometric framework to broader non-commutative settings, linking convolution, moments, and cumulants via Lie groups and Hopf algebras.

Findings

Universal convolution products are represented by pro-unipotent group schemes.

Moment-cumulant relations are interpreted through Lie group exponential and logarithm maps.

The shuffle Hopf algebra plays a universal role in the theory.

Abstract

Here we extend the algebro-geometric approach to free probability, started in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show that any universal convolution product of moments of independent (non)-commutative random variables defined on a graded connected dual semi-group is given by a pro-unipotent group scheme. We show that moment-cumulant formulae have a natural interpretation within the theory of homogeneous Lie groups, which we generalise for the present purpose, and are given by the log and exp map, respectively. Finally, we briefly discuss the universal role of the shuffle Hopf algebra.