Certain actions of finitely generated groups on higher dimensional noncommutative tori

TL;DR

This paper investigates actions of finitely generated abelian groups on higher dimensional noncommutative tori, providing criteria for simplicity and AF algebra structure of the crossed products.

Contribution

It introduces a function to detect when such tori admit specific group actions and characterizes conditions for the crossed product to be an AF algebra.

Findings

A function to determine the existence of specific group actions on noncommutative tori.

Conditions under which the crossed product is an AF algebra.

Characterization of group actions leading to simple noncommutative tori.

Abstract

We study certain actions of finitely generated abelian groups on higher dimensional noncommutative tori. Given a dimension $d$ and a finitely generated abelian group $G$, we apply a certain function to detect whether there is a simple noncommutative $d$-torus which admits a specific action by $G$. We also describe the condition of $G$ under which the associated crossed product is an AF algebra.