Characteristic free Galois rings and generalized Weyl algebras

TL;DR

This paper introduces a new theory of Galois rings and generalized Weyl algebras over arbitrary fields, establishing foundational properties and representation theory, and connecting invariants under complex reflection groups to principal Galois orders.

Contribution

It develops the first comprehensive theory of Galois rings without non-modularity assumptions and introduces infinite rank generalized Weyl algebras with their basic properties.

Findings

Proves an analogue of the Main Theorem for Galois orders over algebraically closed fields.

Develops the theory of infinite rank generalized Weyl algebras and their properties.

Shows invariants of generalized Weyl algebras under certain group actions are principal Galois orders.

Abstract

This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically closed, the analogue of the Main Theorem of the representation theory of Galois orders by V. Futorny and S. Ovsienko. Then we develop a theory of infinite rank generalized Weyl algebras, which was never explicitly introduced in the literature before, and prove its basic properties. We expect their representation theory to be of interest for future works. Finally we show that under very mild assumptions, the invariants of generalized Weyl algebras under the action of non-exceptional irreducible complex reflection groups are a principal Galois orders, greatly generalizing, in an elementary fashion, results obtained previously for the Weyl algebras.