Morita equivalence classes for crossed product of rational rotation algebras

TL;DR

This paper classifies the Morita equivalence classes of crossed products of rational rotation algebras by cyclic subgroups, providing a complete classification for all rational parameters.

Contribution

It extends the classification of crossed products of rotation algebras to rational parameters, completing the Morita equivalence classification for all cases.

Findings

Crossed products with rational $ heta$ are Morita equivalent for the same subgroup.

Complete classification of crossed products $A_ heta times F$ up to Morita equivalence.

Unified understanding of Morita classes for both rational and irrational $ heta$.

Abstract

We study the Morita equivalence classes of crossed products of rotation algebras $A_\theta$, where $\theta$ is a rational number, by finite and infinite cyclic subgroups of $\mathrm{SL}(2, \mathbb{Z})$. We show that for any such subgroup $F$, the crossed products $A_\theta \rtimes F$ and $A_{\theta'} \rtimes F$ are strongly Morita equivalent, where both $\theta$ and $\theta'$ are rational. Combined with previous results for irrational values of $\theta$, our result provides a complete classification of the crossed products $A_\theta \rtimes F$ up to Morita equivalence.