Simple weight modules over weak Generalized Weyl algebras

TL;DR

This paper classifies simple weight modules over weak generalized Weyl algebras of rank one, highlighting the role of endomorphisms in their structure and connecting the results to generalized Heisenberg algebras.

Contribution

It provides a classification framework for simple weight modules over weak generalized Weyl algebras using endomorphism dynamics, extending prior automorphism-based approaches.

Findings

Classification reduces to endomorphism dynamics on maximal ideals

Describes structure of simple weight modules over these algebras

Applications to generalized Heisenberg algebras

Abstract

In this paper we address the problem of classification of simple weight modules over weak generalized Weyl algebras of rank one. The principal difference between weak generalized Weyl algebras and generalized weight algebras is that weak generalized Weyl algebras are defined using an endomorphism rather than an automorphism of a commutative ring $R$. We reduce classification of simple weight modules over weak generalized Weyl algebras to description of the dynamics of the action of the above mentioned endomorphism on the set of maximal ideals. We also describe applications of our results to the study of generalized Heisenberg algebras.