Algebra bundles, projective flatness and rationally-deformed tori
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915505923686400 |
|---|---|
| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the Čech cohomology group $H^2(\mathbb{T},μ_q:=\text{$q^{th}$ roots of unity})$ (respectively $H^2(\mathbb{T},\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))\xrightarrow{\quad}H^2(\mathbb{T},μ(q,\mathcal{C}_{\mathbb{T}}))$ (respectively the first Chern class $c_1(\mathcal{E})$). This is in the spirit of Auslander-Szczarba's result identifying real flat bundles on the torus with their first two Stiefel-Whitney classes, and contrasts with classifying spaces $BΓ$ of compact Lie groups $Γ$ (as opposed to $\mathbb{T}^n\cong B\mathbb{Z}^n$), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori $\mathbb{T}^n_θ$ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to $\mathbb{T}^2_θ$, $θ\in\mathbb{Q}$ by Rieffel and respectively Høegh-Krohn-Skjelbred. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebra bundles, projective flatness and rationally-deformed tori Chirvasitu, Alexandru Algebraic Topology Functional Analysis K-Theory and Homology Operator Algebras Quantum Algebra 55R40, 55R37, 46M20, 16S35, 20J06, 46L85, 54B40, 55R25 We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the Čech cohomology group $H^2(\mathbb{T},μ_q:=\text{$q^{th}$ roots of unity})$ (respectively $H^2(\mathbb{T},\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))\xrightarrow{\quad}H^2(\mathbb{T},μ(q,\mathcal{C}_{\mathbb{T}}))$ (respectively the first Chern class $c_1(\mathcal{E})$). This is in the spirit of Auslander-Szczarba's result identifying real flat bundles on the torus with their first two Stiefel-Whitney classes, and contrasts with classifying spaces $BΓ$ of compact Lie groups $Γ$ (as opposed to $\mathbb{T}^n\cong B\mathbb{Z}^n$), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori $\mathbb{T}^n_θ$ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to $\mathbb{T}^2_θ$, $θ\in\mathbb{Q}$ by Rieffel and respectively Høegh-Krohn-Skjelbred. |
| title | Algebra bundles, projective flatness and rationally-deformed tori |
| topic | Algebraic Topology Functional Analysis K-Theory and Homology Operator Algebras Quantum Algebra 55R40, 55R37, 46M20, 16S35, 20J06, 46L85, 54B40, 55R25 |
| url | https://arxiv.org/abs/2509.10812 |