Algebraic isomorphisms of quantized homogeneous spaces

TL;DR

This paper proves that for certain quantum homogeneous spaces derived from compact semisimple Lie groups, their $q$-deformed function algebras are all mutually non-isomorphic as $*$-algebras for $0<q eq1$.

Contribution

It provides a proof of a folklore theorem showing the non-isomorphism of $q$-deformed algebras for these spaces, clarifying their algebraic distinctions.

Findings

$q$-deformed algebras are mutually non-isomorphic for different $q$

The result applies to homogeneous spaces with Poisson-Lie stabilizers

The proof confirms the algebraic uniqueness of each deformation level

Abstract

We describe a proof of the following folklore theorem: If $\cX = G/K$ is the homogeneous space of a simply connected compact semisimple Lie group with Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions $\CC[\cX_q]$ with $0<q\leq1$ are mutually non-isomorphic as $*$-algebras.