Isomorphisms of quantum spheres

TL;DR

This paper investigates the algebraic structures of quantum spheres and demonstrates that their polynomial deformation algebras are isomorphic only when the deformation parameters are equal, contrasting with their C*-algebraic counterparts.

Contribution

It establishes the precise conditions under which the polynomial algebras of quantum spheres are isomorphic, revealing a finer distinction than previously known.

Findings

Polynomial algebras of quantum spheres are isomorphic only if deformation parameters match.

C*-enveloping algebras are independent of the deformation parameter q.

The isomorphism class of polynomial algebras depends on the deformation parameter q.

Abstract

For $n\in\mathbb{N}$ and $q\in [0,1[$, the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ is described by an associative algebra $\mathcal{A}(S^{2n+1}_q)$ deforming the algebra of polynomial functions on the 2n+1 dimensional unit sphere. Its C*-enveloping algebra is known to be independent of the deformation parameter q. In contrast to what happens in the C*-algebraic setting, we show here that, for all $q,q'$ in the above range, $\mathcal{A}(S^{2n+1}_q)$ is isomorphic to $\mathcal{A}(S^{2n+1}_{q'})$ only if $q=q'$.