Polynomial identities and Azumaya loci for rational quantum spheres

TL;DR

This paper investigates the structure, isomorphisms, and algebraic properties of non-commutative rational quantum spheres, revealing how their invariants and centers relate to classical geometric objects and algebraic finiteness.

Contribution

It provides new results on the isomorphism classes, PI properties, Azumaya loci, and centers of rational quantum spheres, connecting non-commutative geometry with classical topology.

Findings

Isomorphism conditions for $C^*$-algebras of quantum spheres

Recovery of parameters from isomorphism classes

Description of centers as branched covers and their algebraic properties

Abstract

We prove a number of structure and isomorphism results concerning the non-commutative Natsume-Olsen spheres $\mathbb{S}^{2n-1}_{\theta}$ deformed along a skew-symmetric matrix $\theta\in \mathbb{R}$. These include (a) the fact that two $C^*$-algebras of the form $\mathbb{S}^{3}_{\theta}\otimes M_n$ are isomorphic precisely in the obvious cases; (b) the fact that $m$ and $n$ are recoverable from the isomorphism class of $C(\mathbb{S}^{2m-1}_{\theta})\otimes M_n$; (c) the PI character, PI degree and Azumaya loci of $C(\mathbb{S}^{2m-1}_{\theta})$ for rational $\theta$, along with a realization of their centers as (function algebras of) branched cover of $\mathbb{S}^{2n-1}$ and (d) for rational $\theta$ again, the topological finite generation of $C(\mathbb{S}^{2m-1}_{\theta})$ over their centers, with algebraic finite generation equivalent to being classical (equivalently, Azumaya).