Galois Rings, Coulomb branches and the Gelfand-Kirillov Conjecture

TL;DR

This paper investigates the algebraic structure of Galois rings and Coulomb branch algebras, establishing their properties, dimensions, and confirming the Gelfand-Kirillov conjecture for these classes of algebras.

Contribution

It introduces new conditions for Galois rings to be Ore domains and (semi)prime Goldie rings, and proves the Gelfand-Kirillov conjecture for Coulomb branch and related algebras.

Findings

Galois rings can be characterized as Ore domains under certain conditions.

Coulomb branch algebras satisfy the Gelfand-Kirillov conjecture.

Structural properties of affine and double affine Hecke algebras are established.

Abstract

Galois rings and orders, introduced by Futorny and Ovsienko, are embedded into fixed subrings of skew group (or monoid) rings and have many interesting applications to the structure and representation theory of algebras. The paper focuses on their ring theoretical properties which can be deduced from the properties of the associated skew group rings via a localization procedure. In particular, we obtain natural conditions for our rings to be Ore domains and (semi)prime Goldie rings. We also discuss various ring theoretical dimensions and combine powerful theories of Galois rings and PI-rings. Furthermore, we compute dimensions and establish structural properties of spherical Coulomb branch algebras, and show that they verify the Gelfand-Kirillov conjecture. Similar results are obtained for affine and double affine Hecke algebras.