Morita equivalent subalgebras of irrational rotation algebras and real quadratic fields

TL;DR

This paper classifies Morita equivalent subalgebras of irrational rotation algebras, linking their structure to solutions of quadratic Diophantine equations and analyzing their local triviality and index properties.

Contribution

It provides a complete classification of Morita equivalent subalgebras of irrational rotation algebras using quadratic Diophantine equations and characterizes their local triviality and index.

Findings

Classification of Morita equivalent subalgebras

Identification of irrational rotation algebras with locally trivial inclusions

Calculation of indices for these inclusions

Abstract

In this paper, we determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have locally trivial inclusions. We compute the index of the locally trivial inclusions of irrational rotation algebras.