On Morita equivalence for simple Generalized Weyl algebras

TL;DR

This paper investigates Morita equivalence conditions for simple Generalized Weyl algebras, linking algebraic properties to quantum tori isomorphisms and exploring generalizations beyond degree 2 polynomials.

Contribution

It provides a necessary condition for Morita equivalence and reformulates Hodges' results in terms of quantum tori, extending the understanding to higher degrees.

Findings

Established a necessary condition for Morita equivalence.

Reformulated Hodges' result using quantum tori.

Explored potential generalizations for higher degrees.

Abstract

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges' result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for $n\ge 3$.