Liberation theory for noncommutative homogeneous spaces

TL;DR

This paper explores the liberation process in noncommutative homogeneous spaces, classifying certain quantum groups and analyzing quotient spaces with algebraic and probabilistic insights.

Contribution

It introduces a framework for axiomatizing and classifying uniform compact quantum groups and studies their quotient spaces and liberation operations.

Findings

Classification of uniform compact quantum groups.

Analysis of quotient spaces of specific types.

Results on algebraic and probabilistic properties of liberation.

Abstract

We discuss the liberation question, in the homogeneous space setting. Our first series of results concerns the axiomatization and classification of the families of compact quantum groups $G=(G_N)$ which are "uniform", in a suitable sense. We study then the quotient spaces of type $X=(G_M\times G_N)/(G_L\times G_{M-L}\times G_{N-L})$, and the liberation operation for them, with a number of algebraic and probabilistic results.