A geometric approach to noncommutative principal torus bundles

作者：Wagner Stefan^*

来源：Proceedings of the London Mathematical Society, 2013, 106(6): 1179-1222.

DOI：10.1112/plms/pds073

A (smooth) dynamical system with transformation group (n) is a triple (A, (n),alpha), consisting of a unital locally convex algebra A, the n-torus (n) and a group homomorphism alpha:(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 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, (n),alpha) is called a noncommutative principal (n)-bundle, if localization leads to a trivial noncommutative principal (n)-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (nontrivial) noncommutative examples.

• 出版日期2013-6