Finite groups acting on higher dimensional noncommutative tori

Abstract

For the canonical action $\alpha$ of $\operatorname{SL}_2(\mathbb{Z})$ on 2-dimensional simple rotation algebras $\mathcal{A}_\theta$, it is known that if $F$ is a finite subgroup of $\operatorname{SL}_2(\mathbb{Z})$, the crossed products $\mathcal{A}_\theta\rtimes_\alpha F$ are all AF algebras. In this paper we show that this is not the case for higher dimensional noncommutative tori. More precisely, we show that for each $n\geq 3$ there exist noncommutative simple $\phi(n)$-dimensional tori $\mathcal{A}_\Theta$ which admit canonical action of $\mathbb{Z}_n$ and for each odd $n\geq 7$ with $2\phi(n)\geq n+5$ their crossed products $\mathcal{A}_\Theta\rtimes_\alpha \mathbb{Z}_n$ are not AF (with nonzero $K_1$-groups). It is also shown that the only possible canonical action by a finite group on a $3$-dimensional simple torus is the flip action by $\mathbb{Z}_2$. Besides, we discuss the canonical actions by finite groups $\mathbb{Z}_5, \mathbb{Z}_8, \mathbb{Z}_{10}$, and $\mathbb{Z}_{12}$ on the $4$-dimensional torus of the form $\mathcal{A}_\theta\otimes \mathcal{A}_\theta$.