Finite groups acting on higher dimensional noncommutative tori

Abstract

For the canonical action $\alpha$ of ${\mathrm{SL}}_2(\mathbb{Z})$ on 2-dimensional simple rotation algebras ${\mathcal{A}}_{\theta }$, it is known that if $F$ is a finite subgroup of ${\mathrm{SL}}_2(\mathbb{Z})$, the crossed products ${\mathcal{A}}_{\theta }{⋊}_{\alpha }F$ 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 $n\ge 3$ there exist noncommutative simple $\varphi (n)$-dimensional tori ${\mathcal{A}}_{\Theta }$ which admit canonical action of ${\mathbb{Z}}_n$ and for each odd $n\ge 7$ with $2\varphi (n)\ge n+5$ their crossed products ${\mathcal{A}}_{\Theta }{⋊}_{\alpha }{\mathbb{Z}}_n$ 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 ${\mathcal{A}}_{\theta }\otimes {\mathcal{A}}_{\theta }$.

25 pages

Cited by in corpus (1)

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