Noncommutative homogeneous spaces: the matrix case

TL;DR

This paper investigates the structure of noncommutative homogeneous spaces formed from quantum subgroups of unitary groups, analyzing when classical algebraic properties hold and focusing on easy quantum groups.

Contribution

It provides a detailed analysis of the algebraic structure of noncommutative homogeneous spaces, especially in the quantum group and easy quantum group cases, extending classical results.

Findings

Classical Stone-Weierstrass type results do not always hold in the quantum case.

Complete characterization of the dual group case.

Construction and analysis of algebras associated with noncommutative spaces.

Abstract

Given a quantum subgroup $G\subset U_n$ and a number $k\leq n$ we can form the homogeneous space $X=G/(G\cap U_k)$, and it follows from the Stone-Weierstrass theorem that $C(X)$ is the algebra generated by the last $n-k$ rows of coordinates on $G$. In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type $X=G/(G\cap U_k^+)$.