Gelfand-Tsetlin theory for rational Galois algebras

Futorny, Vyacheslav; Grantcharov, Dimitar; Ramirez, Luis Enrique; Zadunaisky, Pablo

doi:10.1007/s11856-020-2048-2

Cited by 11 publications

(17 citation statements)

References 23 publications

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“…Therefore we may apply Theorem 10 to obtain a family of the corresponding modules. In the case when H is a reflection group and satisfies some other conditions (it has to be parabolic with respect to a fixed system of simple roots), the A-module Im(Υ * ), cf Theorem 10, was constructed in [FGRZ,Theorem 7.4]. This recovers the corresponding result of [FGRZ].…”

Section: Applications Of Theorem 4 To Gelfand-zeitlin Modulesmentioning

confidence: 67%

“…The case when A is a rational Galois orders corresponding to any reflection group was later considered in [FGRZ]. In this section, we show how to obtain [RZ,Section 5.6,Theorem], [EMV,Theorem 10] and [FGRZ,Theorem 7.4] using Corollary 6, Theorem 10 and Proposition 14.…”

Section: Applications Of Theorem 4 To Gelfand-zeitlin Modulesmentioning

confidence: 97%

“…Other examples of such algebras are: finite W-algebras of type A and, more general, standard Galois orders of type A, see [FGRZ,Section 8] or [Har] for definition. In Section 5 we will show that standard algebras of type A that preserve the vector space O G are exactly standard Galois orders of type A.…”

Section: Orthogonal Gelfand-zeitlin Algebras Are Examples Of Standard...mentioning

confidence: 99%

“…Gelfand-Zeitlin modules for orthogonal Gelfand-Zeitlin algebras and Galois orders, studied in [EMV,FGRZ,Vi1,Vi2] are examples of Harish-Chandra modules.…”

Section: Harish-chandra Modulesmentioning

confidence: 99%

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Harish–Chandra modules over invariant subalgebras in a skew-group ring

Mazorchuk

Vishnyakova

2021

Asian Journal of Mathematics

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We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV, FGRZ, RZ, Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand-Zeitlin algebras from [EMV] and for rational Galois algebras from [FGRZ]. We also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules.

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Section: Applications Of Theorem 4 To Gelfand-zeitlin Modulesmentioning

confidence: 67%

Section: Applications Of Theorem 4 To Gelfand-zeitlin Modulesmentioning

confidence: 97%

Section: Orthogonal Gelfand-zeitlin Algebras Are Examples Of Standard...mentioning

confidence: 99%

“…Gelfand-Zeitlin modules for orthogonal Gelfand-Zeitlin algebras and Galois orders, studied in [EMV,FGRZ,Vi1,Vi2] are examples of Harish-Chandra modules.…”

Section: Harish-chandra Modulesmentioning

confidence: 99%

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Harish–Chandra modules over invariant subalgebras in a skew-group ring

Mazorchuk

Vishnyakova

2021

Asian Journal of Mathematics

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“…They form a collection of algebras that contains many important examples including: generalized Weyl algebras defined independently by Bavula [Bav92] and Rosenberg [Ros95] in the early nineties, U (gl n ), shifted Yangians and finite W -algebras [FMO10], Coulomb branches [Web19], and U q (gl n ) [FH14]. Their structures and representations have been studied in [Fut+18], [FS18a], [Har20], and [MV18].…”

Section: Galois Ordersmentioning

confidence: 99%

A study of Galois and flag orders

Jauch¹

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Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of gl n (as well as its quantum deformation), generalized Weyl algebras, and shifted Yangians. The main motivation for introducing Galois orders is they provide a setting for studying certain infinite dimensional irreducible representations, called Gelfand-Tsetlin modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois orders with an extra property, which makes them easier to study. All of the mentioned examples are principal Galois orders. In 2019, B. Webster defined principal flag orders which are Morita equivalent to principal Galois orders and further simplifies their study.The purpose of this dissertation is twofold:(1) To introduce a new example of a Galois order, A (gl n ), which is an extension of the enveloping algebra of gl n such that the "Weyl group" of A (gl n ) is the alternating group;(2) To describe some techniques to study such objects including tensor products and morphisms between standard flag orders with conjectured application to the orthogonal Lie algebra.

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Gelfand-Tsetlin Modules: Canonicity and Calculations

Silverthorne,

Webster

2024

Algebr Represent Theor

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Gelfand-Tsetlin theory for rational Galois algebras

Cited by 11 publications

References 23 publications

Harish–Chandra modules over invariant subalgebras in a skew-group ring

Harish–Chandra modules over invariant subalgebras in a skew-group ring

A study of Galois and flag orders

Gelfand-Tsetlin Modules: Canonicity and Calculations

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