Fixed rings of twisted generalized Weyl algebras

TL;DR

This paper investigates the invariants of twisted generalized Weyl algebras (TGWAs) under diagonal automorphisms, showing that fixed rings are often TGWAs themselves and analyzing their algebraic properties and modules.

Contribution

It demonstrates that fixed rings of certain TGWAs under diagonal automorphisms are again TGWAs, expanding understanding of their structure and invariants.

Findings

Fixed rings of ${K}$-finitistic TGWAs of type (A_1)^n and A_2 are again TGWAs.

The fixed rings retain properties like noetherianity and simplicity under certain conditions.

The class of regular, μ-consistent TGWAs is closed under tensor products.

Abstract

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, such as Weyl algebras, primitive quotients of $U(\mathfrak{sl}_2)$, and multiparameter quantized Weyl algebras. In this work, we study invariants of TGWAs under diagonal automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. In particular, this is true for $\Bbbk$-finitistic TGWAs of type $(A_1)^n$ and $A_2$. We apply this theorem to study properties of the fixed ring, such as the noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring. As an auxiliary result, in order to study invariants of tensor products of TGWAs, we prove that the class of regular, $\mu$-consistent TGWAs is closed under tensor products.