Generalized Weyl algebras and diskew polynomial rings

TL;DR

This paper extends generalized Weyl algebras to include rings defined by two endomorphisms, introduces diskew polynomial rings related to GWAs, and provides criteria for their semisimplicity.

Contribution

It introduces diskew polynomial rings as a new class related to GWAs and extends the theory to rings determined by two endomorphisms.

Findings

Introduction of diskew polynomial rings as a new class.

Extension of GWA theory to rings with two endomorphisms.

Semisimplicity criteria for these rings.

Abstract

The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old' GWAs. A new class of rings, the {\em diskew polynomial rings}, is introduced that is closely related to GWAs (they are GWAs under a mild condition). The, so-called, ambiskew polynomial rings are a small subclass of the class of diskew polynomial rings. Semisimplicity criteria are given for generalized Weyl algebras and diskew polynomial rings.