Polynomial identities and Azumaya loci for rational quantum spheres

Alexandru Chirvasitu

Vol. 342 (2026), No. 1, 37–62

DOI: 10.2140/pjm.2026.342.37

Abstract

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

Keywords

Azumaya algebra, Chern class, PI algebra, bundle, classifying space, noncommutative sphere, noncommutative torus, projectively flat

Mathematical Subject Classification

Primary: 46L52, 16H05, 46M20, 55R25, 55R37, 46L85, 16R10, 55R40

Milestones

Received: 6 August 2025

Accepted: 18 January 2026

Published: 23 March 2026