A geometric approach to 1-singular Gelfand–Tsetlin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">gl</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>-modules

Vishnyakova, Elizaveta

doi:10.1016/j.difgeo.2017.11.008

Cited by 19 publications

(14 citation statements)

References 3 publications

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“…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%

“…In the present paper we define and study a simultaneous geometric generalization of orthogonal Gelfand-Zeitlin algebras and Galois algebras. Both our construction and methods of study are inspired by the geometric approach of [Vi1,Vi2] to singular Gelfand-Zeitlin modules and is formulated in elementary sheaf-theoretic terms. To any semidirect product G V of a finite group G and a complex-analytic or linear algebraic group V , we associate the corresponding skew-group ring S, see Subsection 2.1 for a precise definition.…”

Section: Introductionmentioning

confidence: 99%

“…In a special case which we call standard algebras of type A, we give an explicit description of our algebras as subalgebras in the universal ring as introduced in [Vi2]. Our geometric approach also naturally provides a construction of a large family of (simple) modules over our algebras, generalizing [Vi1,Vi2,EMV]. We note that, the general case of our construction seems to be outside the scope of Harish-Chandra subalgebras and Gelfand-Zeitlin modules as defined in [DFO].…”

Section: Introductionmentioning

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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“…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%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Mazorchuk

Vishnyakova

2021

Asian Journal of Mathematics

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“…Next step was to consider 1-singular case when there is just one pair in only one row with integer difference. A break through was a paper [7] (see also [8], [25], [23]) where such modules were explicitly constructed, followed by general constructions of certain "universal" modules in [22] and [24].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial construction of Gelfand-Tsetlin modules for $\mathfrak{gl}_n$

Futorny¹,

Ramirez²,

Zhang³

2016

Preprint

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We propose a new effective method of constructing explicitly Gelfand -Tsetlin modules for gl n . We obtain a large family of simple modules that have a basis consisting of Gelfand-Tsetlin tableaux, the action of the Lie algebra is given by the Gelfand-Tsetlin formulas and with all Gelfand-Tsetlin multiplicities equal 1. As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev's continuation construction to define a module which was conjectured by Lemire and Patera.

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“…Further, infinite dimensional GelfandTsetlin modules for gl n were studied in [18], [25], [26], [3], [31], [7], [27], [28], [30], [8], [9], [10], [11], [12], [37], [32], [35], [36] among the others. These representations have close connections to different concepts in Mathematics and Physics (cf.…”

Section: Introductionmentioning

confidence: 99%

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

2018

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Abstract. The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for Uq(gl n ). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand-Tsetlin multiplicities bounded by 1 [16], [17]. In particular, our method works for q = 1 providing new families of irreducible Gelfand-Tsetlin modules for gl n . This generalizes the results of [10] and [15].

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A geometric approach to 1-singular Gelfand–Tsetlin gln-modules

Cited by 19 publications

References 3 publications

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

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

Combinatorial construction of Gelfand-Tsetlin modules for $\mathfrak{gl}_n$

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

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