A Geometric Approach to Noncommutative Principal Torus Bundles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2011
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| _version_ | 1866908728909889536 |
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| author | Wagner, Stefan |
| author_facet | Wagner, Stefan |
| contents | A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal torus bundles based on such dynamical systems. Our approach is inspired by the classical setting: In fact, after recalling the definition of a trivial noncommutative principal torus bundle, we introduce a convenient (smooth) localization method for noncommutative algebras and say that a dynamical system $(A,\mathbb{T}^n,α)$ is called a noncommutative principal $\mathbb{T}^n$-bundle, if localization leads to a trivial noncommutative principal $\mathbb{T}^n$-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (non-trivial) noncommutative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1108_4294 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | A Geometric Approach to Noncommutative Principal Torus Bundles Wagner, Stefan Differential Geometry Mathematical Physics 46L87, 55R10 (primary), 37B05, 17A60 (secondary) A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal torus bundles based on such dynamical systems. Our approach is inspired by the classical setting: In fact, after recalling the definition of a trivial noncommutative principal torus bundle, we introduce a convenient (smooth) localization method for noncommutative algebras and say that a dynamical system $(A,\mathbb{T}^n,α)$ is called a noncommutative principal $\mathbb{T}^n$-bundle, if localization leads to a trivial noncommutative principal $\mathbb{T}^n$-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (non-trivial) noncommutative examples. |
| title | A Geometric Approach to Noncommutative Principal Torus Bundles |
| topic | Differential Geometry Mathematical Physics 46L87, 55R10 (primary), 37B05, 17A60 (secondary) |
| url | https://arxiv.org/abs/1108.4294 |