Trivial noncommutative principal torus bundles

Stefan Wagner. "Trivial noncommutative principal torus bundles." Banach Center Publications 96.1 (2011): 299-317. <http://eudml.org/doc/286120>.

@article{StefanWagner2011,
abstract = {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.},
author = {Stefan Wagner},
journal = {Banach Center Publications},
keywords = {dynamical systems; (trivial) principal -bundles; (trivial) noncommutative principal -bundles; -graded algebras; factor systems; noncommutative tori},
language = {eng},
number = {1},
pages = {299-317},
title = {Trivial noncommutative principal torus bundles},
url = {http://eudml.org/doc/286120},
volume = {96},
year = {2011},
}

TY - JOUR
AU - Stefan Wagner
TI - Trivial noncommutative principal torus bundles
JO - Banach Center Publications
PY - 2011
VL - 96
IS - 1
SP - 299
EP - 317
AB - 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.
LA - eng
KW - dynamical systems; (trivial) principal -bundles; (trivial) noncommutative principal -bundles; -graded algebras; factor systems; noncommutative tori
UR - http://eudml.org/doc/286120
ER -