Galois Rings, Coulomb branches and the Gelfand-Kirillov Conjecture
Vyacheslav Futorny, Jonas T. Hartwig, Erich C. Jauch, João Schwarz
TL;DR
The paper develops a localization framework for Galois rings and orders, enabling the transfer of structural properties from ambient skew group rings to fixed subrings. It applies these methods to spherical Coulomb branch algebras and to affine/double affine Hecke algebras, proving that these families satisfy GK-type phenomena and detailing their centers and over-center structures. It then analyzes growth and Krull dimensions, establishing LD-stability and precise invariants (GK, Tdeg, Krull) for Coulomb branches and related algebras, while classifying PI versus non-PI Galois rings with explicit examples. Collectively, the results provide a robust noncommutative framework for understanding GK-type conjectures, centers, and dimensions across Coulomb branches, DAHA, and related algebraic objects, with broad implications for representation theory and invariant theory.
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.