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2011 | 96 | 1 | 299-317
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Trivial noncommutative principal torus bundles

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A (smooth) dynamical system with transformation group 𝕋ⁿ is a triple (A,𝕋ⁿ,α), consisting of a unital locally convex algebra A, the n-torus 𝕋ⁿ and a group homomorphism α: 𝕋ⁿ → Aut(A), which induces a (smooth) continuous action of 𝕋ⁿ on A. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of trivial principal 𝕋ⁿ-bundles based on such dynamical systems, i.e., we call a dynamical system (A,𝕋ⁿ,α) a trivial noncommutative principal 𝕋ⁿ-bundle if each isotypic component contains an invertible element. Each trivial principal bundle (P,M,𝕋ⁿ,q,σ) gives rise to a smooth trivial noncommutative principal 𝕋ⁿ-bundle of the form $(C^{∞}(P),𝕋ⁿ,α)$. Conversely, if P is a manifold and $(C^{∞}(P),𝕋ⁿ,α)$ a smooth trivial noncommutative principal 𝕋ⁿ-bundle, then we recover a trivial principal 𝕋ⁿ-bundle. While in classical (commutative) differential geometry there exists up to isomorphy only one trivial principal 𝕋ⁿ-bundle over a given manifold M, we will see that the situation completely changes in the noncommutative world. Moreover, it turns out that each trivial noncommutative principal 𝕋ⁿ-bundle possesses an underlying algebraic structure of a ℤⁿ-graded unital associative algebra, which might be thought of an algebraic counterpart of a trivial principal 𝕋ⁿ-bundle. In the second part of this paper we provide a complete classification of this underlying algebraic structure, i.e., we classify all possible trivial noncommutative principal 𝕋ⁿ-bundles up to completion.
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  • Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
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bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-22
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