Irreducible Generic Gelfand-Tsetlin Modules of gl(n)

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

doi:10.3842/sigma.2015.018

Cited by 24 publications

(18 citation statements)

References 28 publications

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“…As a non-trivial example, we discuss the structure of generic modules of the gl(N ) Kac-Moody algebra and corresponding W-algebras. We conjecture that a subclass of modules coming from GW defects can be identified with modules induced from generic Gelfand-Tsetlin modules of gl(N ) from [41] and their Walgebra analogues. Generally, one obtains wild classes of irregular modules [42,43].…”

Section: Jhep05(2019)159mentioning

confidence: 97%

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

Procházka¹,

Rapčák

2019

J. High Energ. Phys.

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We study the structure of modules of corner vertex operator algebras arrising at junctions of interfaces in N = 4 SYM. In most of the paper, we concentrate on truncations of W 1+∞ associated to the simplest trivalent junction. First, we generalize the Miura transformation for W N 1 to a general truncation Y N 1 ,N 2 ,N 3. Secondly, we propose a simple parametrization of their generic modules, generalizing the Yangian generating function of highest weight charges. Parameters of the generating function can be identified with exponents of vertex operators in the free field realization and parameters associated to Gukov-Witten defects in the gauge theory picture. Finally, we discuss some aspect of degenerate modules. In the last section, we sketch how to glue generic modules to produce modules of more complicated algebras. Many properties of vertex operator algebras and their modules have a simple gauge theoretical interpretation.

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Section: Jhep05(2019)159mentioning

confidence: 97%

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

Procházka¹,

Rapčák

2019

J. High Energ. Phys.

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“…Let F N C (M ) = V . Since any maximal ideal m ′ in the Gelfand-Tsetlin support of M is generic we have dim M m ′ = 1, implying dim V m ′ = 1[7]. Moreover, M is a unique Gelfand-Tsetlin module that has m in its Gelfand-Tsetlin support.…”

mentioning

confidence: 98%

Drinfeld category and the classification of singular Gelfand–Tsetlin $\displaystyle \mathfrak{gl}_n$-modules

Futorny

Grantcharov

Ramirez

2017

International Mathematics Research Notices

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We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand-Tsetlin modules with 1-singularity. An explicit construction of such modules was given in [8]. In particular, we show that the modules constructed in [8] exhaust all irreducible Gelfand-Tsetlin modules with 1-singularity. To prove the result we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y (gl n ) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.1991 Mathematics Subject Classification. Primary 17B67.Henceforth, we define V (L) to be the complex vector space with basis the set L Z , i.e.An open problem which dates back to the work of Gelfand and Graev, [11], is to define a gl n -module structure on a subspace of V (L) annihilated by some maximal ideal of the Gelfand-Tsetlin subalgebra. Solutions to this problem are known in various cases, the most fundamental of which relies on the construction of Gelfand-Tsetlin bases of finite dimensional representations of g.Another important solution concerns the generic Gelfand-Tsetlin modules. More precisely, a pair of entries (l mi , l mj ) of L such that l mi − l mj ∈ Z is called a singular pair. If T (L) contains a singular pair in the m-th row for some 2 ≤ m ≤ n − 1 then we call T (L) a singular tableau (and L a singular element in T n (C)). A tableau T (L) is 1-singular if it contains exactly one singular pair. If T (L) is not singular it is called generic. For a generic tableau T (L), one can imitate the construction of Gelfand-Tsetlin bases of finite-dimensional representations and construct a gl nmodule structure on V (L), see for example [5].The study of singular modules V (L) was initiated in [8], where a module structure on V (L) was introduced for any 1-singular tableau L. The latter construction was generalized in [9] for singular tableaux L with multiple singular pairs such that the difference of the entries of any two distinct singular pairs is noniteger. By taking irreducible quotients of V (L) new irreducible Gelfand-Tsetlin modules for gl n were constructed. In particular, understanding the 1-singular case helped us to complete classification of irreducible Gelfand-Tsetlin gl 3 -modules in [10]. Explicit basis for a class of irreducible 1-singular Gelfand-Tsetlin gl n -modules was constructed in [13].We note that to each L ∈ T n (C) we associate the maximal ideal m L of the Gelfand-Tsetlin subalgebra Γ generated by c ij −γ ij (L), 1 ≤ j ≤ i ≤ n, where c ij are the generators of Γ, and γ ij are symmetric polynomials defined in (3). Furthermore, for every maximal ideal m of Γ there exists an irreducible Gelfand-Tsetlin module V such that V m = 0 (see (2) for the definition of V m ) and the number of such non-isomorphic irreducible modules V is finite, [24]. In fact, for a generic L there exists a unique up to isomorphism irreducible module V such that V mL = 0. On the othe...

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“…We expect the representation theory associated to more general toric Calabi-Yau 3-folds to be more complicated. [88] conjectured appearance of modules induced from generic Gelfand-Tsetlin modules of [27] and various irregular modules of [41,43]. 12 For simplicity, we again decouple W 1 as in the Virasoro case above.…”

Section: Hyperbolic Localizationmentioning

confidence: 99%

Cohomological Hall Algebras, Vertex Algebras and Instantons

Rapčák

Soibelman

Yang

et al. 2019

Commun. Math. Phys.

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We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr 1 ,r 2 ,r 3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau 3-folds.

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Irreducible Generic Gelfand-Tsetlin Modules of gl(n)

Abstract: Abstract. We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra gl(n).

Cited by 24 publications

References 28 publications

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

Drinfeld category and the classification of singular Gelfand–Tsetlin $\displaystyle \mathfrak{gl}_n$-modules

Cohomological Hall Algebras, Vertex Algebras and Instantons

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