Generalized Verma Modules Induced from sl(2,C) and Associated Verma Modules

Khomenko, Oleksandr; Mazorchuk, Volodymyr

doi:10.1006/jabr.2001.8815

Cited by 12 publications

(7 citation statements)

References 18 publications

(33 reference statements)

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“…In particular this generalizes the results of D. Miličić and W. Soergel [MS] for Whittaker modules, the results of V. Mazorchuk and S. Ovsienko [MO] for modules induced from generic Gelfand-Zetlin modules and the results of V. Mazorchuk and the author [KM3,KM4] for the modules induced from dense sl(2)-modules.…”

Section: Structure Of Induced Modulesmentioning

confidence: 52%

Categories with projective functors

Khomenko

2005

Proc. London Math. Soc.

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We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors. 2000 Mathematics Subject Classification 17B10.

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Section: Structure Of Induced Modulesmentioning

confidence: 52%

Categories with projective functors

Khomenko

2005

Proc. London Math. Soc.

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“…Finally, we would like to compare our results with those obtained in [KM3] in the case a = sl(2, C). In the present paper we extend those results, for instance, by partial description of the multiplicities of simple subquotients in a GVM.…”

Section: Then the Category Of Modules Presentable By F ⊗ V F Finitementioning

confidence: 80%

“…The generalized Verma modules induced from finite-dimensional modules were studied by Jantzen in [J1] and by Rocha-Caridi in [R]. The generalized Verma modules induced from infinite-dimensional weight modules, in particular, from Gelfand-Zetlin modules, were studied by Futorny, Khomenko, Mazorchuk and Ovsienko in [FM,KM1,KM2,KM3,MO] (see also references therein). The generalized Verma modules induced from Whittaker modules were studied by McDowell in [Mc1,Mc2], by Miličić and Soergel in [MS1,MS2] and by Backelin in [Ba].…”

Section: Introductionmentioning

confidence: 99%

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Structure of Modules Induced from Simple Modules with Minimal Annihilator

Khomenko

Mazorchuk

2004

Can. j. math.

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Abstract. We study the structure of generalized Verma modules over a semi-simple complex finitedimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-GelfandGelfand category O and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

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“…The question about the structure of submodules of a Verma module arose in the original paper of Verma [5]. As a natural generalization of Verma modules, the generalized Verma modules are modules induced, starting from arbitrary simple modules (not necessarily finite-dimensional), from a parabolic subalgebra and a complex semisimple Lie algebra (see [6][7][8][9]). One of the main questions about generalized Verma modules is their structure, i.e., reducibility, submodules, equivalence, etc.…”

Section: Introductionmentioning

confidence: 99%

Some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras

Zheng

Zhang

2017

Czech.Math.J.

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This paper is primarily concerned with generalized reduced Verma modules over Zgraded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of Z-graded modular Lie superalgebras of Cartan type. (Y. Zhang).Let V be an L-module. The vector space V * := Hom F (V, F) obtains the structure of an L-module by means of (We consider the subalgebra K := L 0 ⊕ L + of a Z-graded Lie superalgebra L = ⊕ i∈Z L i . Let {e 1 , . . . , e k } be a basis of L − ∩ L0 and {ξ 1 , . . . , ξ l } be a basis of L − ∩ L1. As L − ∩ L0 operates on L by nilpotent transformation, there exist m i ∈ N 0 , 1 ≤ i ≤ k such that z i := e p m i i ∈ U (L − ) ∩ Z(U (L)), 1 ≤ i ≤ k,

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Generalized Verma Modules Induced from sl(2,C) and Associated Verma Modules

Cited by 12 publications

References 18 publications

Categories with projective functors

Categories with projective functors

Structure of Modules Induced from Simple Modules with Minimal Annihilator

Some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras

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