Harish-Chandra Subalgebras and Gelfand-Zetlin Modules

Drozd, Yuriy A.; Futorny, Vyacheslav; Ovsienko, Serge

doi:10.1007/978-94-017-1556-0_5

Cited by 75 publications

(101 citation statements)

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“…In the case of the one-column pyramids π , this reduces to the definition of the Gelfand-Tsetlin modules for gl m [9]. Note also that the admissible W (π)-modules of [8] are Gelfand-Tsetlin modules.…”

Section: The Gelfand-kirillov Problem For W (π): Does D(w (π)) D Km mentioning

confidence: 95%

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The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

Futorny

Molev

Ovsienko

2010

Advances in Mathematics

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We address two problems with the structure and representation theory of finite W -algebras associated with general linear Lie algebras. Finite W -algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W -algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W -algebras.

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Section: The Gelfand-kirillov Problem For W (π): Does D(w (π)) D Km mentioning

confidence: 95%

“…Recall that a commutative subalgebra A of an associative algebra B is called a HarishChandra subalgebra if for any b ∈ B, the A-bimodule AbA if finitely generated both as a left and as a right A-module [10].…”

Section: Commutesmentioning

confidence: 99%

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

Futorny

Molev

Ovsienko

2010

Advances in Mathematics

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“…The definition and properties of (extended) OGZ algebras are closely related to those of generic Gelfand-Zetlin ᒄᒉ(n, C)-modules ( [3]). …”

Section: (Extended) Ogz-algebrasmentioning

confidence: 99%

“…This implies that M(ᒊ) is a simpleÂ-module. In the case g(ᒊ) = ᒊ for any 0 = g ∈ Z n its uniqueness follows easily from general nonsense (see [3,Theorem 18] [6], the later being associated with a biserial graph. In the present paper and in Corollary 6.2 we have removed this restriction.…”

Section: Application Of the Shapovalov Form To Construction Of Simplementioning

confidence: 99%

Some associative algebras related to $U(\mathfrak g)$ and twisted generalized Weyl algebras

Mazorchuk¹,

Ponomarenko²,

Turowska³

2003

MATH. SCAND.

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We prove that both Mickelsson step algebras and Orthogonal Gelfand-Zetlin algebras are twisted generalized Weyl algebras. Using an analogue of the Shapovalov form we construct all weight simple graded modules and some classes of simple weight modules over a twisted generalized Weyl algebra, improving the results from [6], where a particular class of algebras was considered and only special modules were classified.

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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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Harish-Chandra Subalgebras and Gelfand-Zetlin Modules

Cited by 75 publications

References 0 publications

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

Some associative algebras related to $U(\mathfrak g)$ and twisted generalized Weyl algebras

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

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