Certain actions of finitely generated abelian groups on higher dimensional noncommutative tori

Zhuofeng He  (University of Tokyo)

9:30 am to 10:30 am, June 9th, 2015    Science Building A1510

Abstract:

We study certain actions of finitely generated abelian groups on higher dimensional noncommutative tori through this paper. 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 $G$. Such an action is a natural generalization of actions of finite subgroups of ${\rm SL}_2(\mathbb{Z})$ on 2-torus $\mathbb{T}\cong \mathbb{R}^2/\mathbb{Z}^2$. For possible cases, we provide a way to construct each such action of $G$ on a simple higher dimensional noncommutative torus, and then compute the $K$-theory of the resulting crossed product. We also describe a sufficient and necessary condition of $G$ under which the associated crossed product is an AF algebra.