Fixed rings of quantum generalized Weyl algebras

TL;DR

This paper investigates the structure of fixed rings of quantum generalized Weyl algebras under finite automorphisms, extending existing theorems and analyzing properties like global dimension and simplicity.

Contribution

It extends a theorem of Jordan and Wells to quantum GWAs and characterizes their fixed rings under diagonal automorphisms, including properties such as rigidity and simplicity.

Findings

Determined the fixed rings of quantum GWAs under diagonal automorphisms.

Analyzed properties like global dimension, rigidity, and simplicity of these fixed rings.

Extended classical theorems to the quantum setting.

Abstract

Generalized Weyl Algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings, including global dimension, rigidity, and simplicity.