Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $SL_2(\mathbb{Z})$

TL;DR

This paper classifies when crossed products of irrational rotation algebras by cyclic subgroups of SL_2(Z) are isomorphic or Morita equivalent, based on K-theory and matrix invariants.

Contribution

It provides explicit criteria for isomorphism and Morita equivalence of these crossed products, extending understanding of their classification via K-theory and matrix invariants.

Findings

Isomorphism characterized by theta and matrix equivalence of I-A^{-1} and I-B^{-1}.

Morita equivalence characterized by GL_2(Z) orbit of theta and matrix invariance.

Determined Morita classes for crossed products with finite subgroups of SL_2(Z).

Abstract

Let $\theta, \theta'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_{\theta}\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_{\theta} \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{\theta'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $\theta = \pm\theta' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_{\theta} \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{\theta'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $\theta$ and $\theta'$ are in the same $GL_2(\mathbb{Z})$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of $\mathcal{A}_{\theta} \rtimes F$ for any finite subgroup $F$ of $SL_2(\mathbb{Z})$.