1 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet2 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet
A (smooth) dynamical system with transformation group 핋n is a triple (A, 핋n,α), consisting of a unital locally convex algebra A, the n-torus 핋n and a group homomorphism α:핋n→Aut(A), which induces a (smooth) continuous action of 핋n 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, 핋n,α) is called a noncommutative principal 핋n-bundle, if localization leads to a trivial noncommutative principal 핋n-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (nontrivial) noncommutative examples.
London Mathematical Society. Proceedings, 2013, Vol 106, Issue 6, p. 1179-1222