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Characteristic free Galois rings and generalized Weyl algebras

Joao Schwarz

TL;DR

The paper develops a characteristic-free theory of Galois rings and orders, removing the non-modularity restriction and extending the Main Theorem of Galois order theory to modular prime characteristic. It introduces infinite rank generalized Weyl algebras via ordinal-indexed direct limits, establishes simplicity and Ore-domain criteria, and analyzes their fixed rings under reflection groups, proving these fixed rings are principal Galois orders. By providing both constructive frameworks and key counterexamples, the work broadens the landscape of noncommutative invariant theory and paves the way for a deeper representation theory of Galois orders and infinite rank GWAs. The results have potential implications for Harish-Chandra modules, skew monoid rings, and invariant theory in noncommutative algebra, with concrete applications to tensorial GWAs and fixed-ring phenomena under finite group actions.

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.

Characteristic free Galois rings and generalized Weyl algebras

TL;DR

The paper develops a characteristic-free theory of Galois rings and orders, removing the non-modularity restriction and extending the Main Theorem of Galois order theory to modular prime characteristic. It introduces infinite rank generalized Weyl algebras via ordinal-indexed direct limits, establishes simplicity and Ore-domain criteria, and analyzes their fixed rings under reflection groups, proving these fixed rings are principal Galois orders. By providing both constructive frameworks and key counterexamples, the work broadens the landscape of noncommutative invariant theory and paves the way for a deeper representation theory of Galois orders and infinite rank GWAs. The results have potential implications for Harish-Chandra modules, skew monoid rings, and invariant theory in noncommutative algebra, with concrete applications to tensorial GWAs and fixed-ring phenomena under finite group actions.

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.
Paper Structure (10 sections, 59 theorems, 9 equations)

This paper contains 10 sections, 59 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reminder of the DFO setting
  4. Inductive limit of rings
  5. Somme noncommutative invariant theory
  6. Characteristic free Galois rings
  7. Main theorem of Galois orders in prime characteristic
  8. Examples and counter-examples
  9. Generalized Weyl algebras and their infinite rank analogues
  10. Fixed rings of generalized Weyl algebras

Key Result

Theorem 2.4

Let $\Gamma \subset \Lambda$ be two associative algebras, with $\Gamma$ Noetherian and $\Lambda$ a finitely generated left and righ $\Gamma$-module. Let $A$ be an associative algebra containing a localization of $\Lambda$, called $L$; and generated, as an algebra, by $L$ and $\mathsf{X}$, where each

Theorems & Definitions (132)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 122 more