A Geometric Approach to Noncommutative Principal Torus Bundles

TL;DR

This paper introduces a new geometric framework for understanding noncommutative principal torus bundles using dynamical systems, extending classical concepts and providing novel examples.

Contribution

It develops a localization-based approach to characterize noncommutative principal torus bundles, bridging classical and noncommutative geometry.

Findings

Defines noncommutative principal $\

Provides a localization method for noncommutative algebras

Extends classical principal bundle theory to noncommutative setting

Abstract

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,\alpha)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $\alpha:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ 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 $(A,\mathbb{T}^n,\alpha)$ is called a noncommutative principal $\mathbb{T}^n$-bundle, if localization leads to a trivial noncommutative principal $\mathbb{T}^n$-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (non-trivial) noncommutative examples.