Fixed rings of generalized Weyl algebras

TL;DR

This paper investigates the structure of fixed rings under automorphisms in generalized Weyl algebras, proving they remain GWAs in certain cases and extending classical results to this setting.

Contribution

It demonstrates that fixed rings of finite cyclic automorphisms are GWAs for degree two polynomials and extends Auslander's theorem to these automorphisms.

Findings

Fixed rings are GWAs for degree two polynomials.

Extension of Auslander's theorem to cyclic automorphisms of GWAs.

Partial results for higher degree polynomials.

Abstract

We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical GWA, extending a result of Jordan and Wells. Partial results are provided for the case of higher degree polynomials. In addition, we establish a version of Auslander's theorem for finite cyclic groups of filtered automorphisms acting on classical GWAs.