Polynomial identities and Azumaya loci for rational quantum spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915433225912320 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | We prove a number of structure and isomorphism results concerning the non-commutative Natsume-Olsen spheres $\mathbb{S}^{2n-1}_θ$ deformed along a skew-symmetric matrix $θ\in \mathbb{R}$. These include (a) the fact that two $C^*$-algebras of the form $\mathbb{S}^{3}_θ\otimes M_n$ are isomorphic precisely in the obvious cases; (b) the fact that $m$ and $n$ are recoverable from the isomorphism class of $C(\mathbb{S}^{2m-1}_θ)\otimes M_n$; (c) the PI character, PI degree and Azumaya loci of $C(\mathbb{S}^{2m-1}_θ)$ 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 $C(\mathbb{S}^{2m-1}_θ)$ over their centers, with algebraic finite generation equivalent to being classical (equivalently, Azumaya). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial identities and Azumaya loci for rational quantum spheres Chirvasitu, Alexandru Quantum Algebra Functional Analysis Operator Algebras Rings and Algebras 46L52, 16H05, 46M20, 55R25, 55R37, 46L85, 16R10, 55R40 We prove a number of structure and isomorphism results concerning the non-commutative Natsume-Olsen spheres $\mathbb{S}^{2n-1}_θ$ deformed along a skew-symmetric matrix $θ\in \mathbb{R}$. These include (a) the fact that two $C^*$-algebras of the form $\mathbb{S}^{3}_θ\otimes M_n$ are isomorphic precisely in the obvious cases; (b) the fact that $m$ and $n$ are recoverable from the isomorphism class of $C(\mathbb{S}^{2m-1}_θ)\otimes M_n$; (c) the PI character, PI degree and Azumaya loci of $C(\mathbb{S}^{2m-1}_θ)$ 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 $C(\mathbb{S}^{2m-1}_θ)$ over their centers, with algebraic finite generation equivalent to being classical (equivalently, Azumaya). |
| title | Polynomial identities and Azumaya loci for rational quantum spheres |
| topic | Quantum Algebra Functional Analysis Operator Algebras Rings and Algebras 46L52, 16H05, 46M20, 55R25, 55R37, 46L85, 16R10, 55R40 |
| url | https://arxiv.org/abs/2508.04922 |