A geometric approach to noncommutative principal torus bundles. Issue 6 (4th January 2013)

• Record Type: Journal Article
• Authors: Wagner, Stefan
• Is Part Of: Proceedings of the London Mathematical Society. Volume 106:Issue 6(2012:Jun.)
• Page Start: 1179
• Page End: 1222
• Publication Date: 2013-01-04
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pds073
• Languages: English
• ISSNs: 0024-6115

Abstract

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