Trivial noncommutative principal torus bundles

Stefan Wagner

Banach Center Publications, Tome 95 (2011), p. 299-317 / Harvested from The Polish Digital Mathematics Library

Résumé

A (smooth) dynamical system with transformation group ⁿ is a triple (A,ⁿ,α), consisting of a unital locally convex algebra A, the n-torus ⁿ and a group homomorphism α: ⁿ → Aut(A), which induces a (smooth) continuous action of ⁿ on A. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of trivial principal ⁿ-bundles based on such dynamical systems, i.e., we call a dynamical system (A,ⁿ,α) a trivial noncommutative principal ⁿ-bundle if each isotypic component contains an invertible element. Each trivial principal bundle (P,M,ⁿ,q,σ) gives rise to a smooth trivial noncommutative principal ⁿ-bundle of the form $(C^{∞}(P),ⁿ,α)$. Conversely, if P is a manifold and $(C^{∞}(P),ⁿ,α)$ a smooth trivial noncommutative principal ⁿ-bundle, then we recover a trivial principal ⁿ-bundle. While in classical (commutative) differential geometry there exists up to isomorphy only one trivial principal ⁿ-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 ⁿ-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 ⁿ-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 ⁿ-bundles up to completion.

Détail

Publié le : 2011-01-01

Zbl 1261.46066

EUDML-ID : urn:eudml:doc:286120