Galois Rings, Coulomb branches and the Gelfand-Kirillov Conjecture

2025/07/14 by Vyacheslav Futorny, Futorny, Vyacheslav, Jonas T. Hartwig +5
Mathematics · #16E65 16H10 16N60 16P90 16P50 16P60 16R99 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2507.10782

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

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.

Citations

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• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
• Noncommutative Noether's problem for complex reflection groups
• Homological Transcendence Degree
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• Residue construction of Hecke algebras

Related

• Gelfand-Tsetlin modules in the Coulomb context
• Some open problems in the context of skew PBW extensions and semi-graded\n rings
• Some homological properties of skew PBW extensions
• Ore and Goldie theorems for skew PBW extensions
• Gelfand-Kirillov dimension for rings