Polynomial identities and Azumaya loci for rational quantum spheres

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).