Simple supermodules over Lie superalgebras

Chen, Chih-Whi; Mazorchuk, Volodymyr

doi:10.1090/tran/8303

Cited by 20 publications

(35 citation statements)

References 27 publications

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“…Secondly, we generalize results of [14] and we use these results along with the classification of simple bounded modules over M(C × , g) to construct a family of level zero simple bounded modules over g. In the last subsection we assume that g is a basic classical Lie superalgebras of type I in order to define analogs of Kac g-modules. The main result of this subsection generalizes to the affine setting the reduction results from [19] and [11] for bounded modules.…”

Section: Introductionmentioning

confidence: 64%

“…Following [11], for a simple weight g 0 -module S we define the Kac module associated to S to be the induced g-module K(S) := U( g) ⊗ U ( g 0 ⊕ g 1 ) S, where we are assuming that g 1 S = 0. It is easy to prove that K(S) admits a unique simple quotient which we denote by V (S).…”

Section: 3mentioning

confidence: 99%

“…in the case of sl(n|m). This phenomenon was studied in [11] for general superalgebras with finite-dimensional odd part, in which case simple modules are obtained as simple quotients of Kac modules. For basic classical Lie superalgebras of type I, we use the Kac functor to reduce the classification of all bounded simple g-modules of level zero to the classification of the same class of modules over g 0 .…”

Section: Introductionmentioning

confidence: 99%

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Classification of simple Harish-Chandra modules for basic classical map superalgebras

Calixto,

Futorny,

Rocha

2021

Preprint

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We obtain a classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolic induced from a simple cuspidal bounded module over a cuspidal map superalgebra. Further on, any simple cuspidal bounded module is isomorphic to an evaluation module. As an application, we obtain a classification of all simple Harish-Chandra modules for basic classical loop superalgebras. Extending these results to affine Kac-Moody Lie superalgebras obtained by adding the degree derivation, we construct a family of bounded simple modules of level zero, and conjecture that all bounded simple cuspidal modules belong to this family. Finally, we show that for affine Kac-Moody Lie superalgebras of type I the Kac induction functor reduces the classification of all simple bounded modules to the classification of the same class of modules over the even part, whose classification is claimed by Dimitrov and Grantcharov.

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Section: Introductionmentioning

confidence: 64%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of simple Harish-Chandra modules for basic classical map superalgebras

Calixto,

Futorny,

Rocha

2021

Preprint

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“…Recall that S denotes the parity shift functor. It turns out that, for any simple g0-module V , we have (see also [CM,Corollary 4.5]):…”

Section: A (Quasireductive) Lie Superalgebra G Is Referred To As a Li...mentioning

confidence: 99%

Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras

Chen¹

2021

Preprint

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We extend Kostant's result on annihilator ideals of non-singular simple Whittaker modules over Lie algebras to (possibly singular) simple Whittaker modules over Lie superalgebras. We describe these annihilator ideals in terms of certain primitive ideals coming from the category O for quasireductive Lie superalgebras.To determine these annihilator ideals, we develop annihilator-preserving equivalences between certain full subcategories of the Whittaker category N and the categories of certain projectively presentable modules in the category O. These equivalences lead to a classification of simple Whittaker modules that lie in the integral central blocks when restricted to the even subalgebra. We make a connection between the linkage classes of integral blocks of O and of N . In particular, they can be computed via Kazhdan-Lusztig combinatorics for Lie superalgebras of type gl and osp. We then give a description of the integral blocks of the category N of Whittaker modules for Lie superalgebras gl(m|n), osp(2|2n) and pe(n). Contents 1. Introduction 1 2. Preliminaries 8 3. Realizations of completion and twisting functors 14 4. The category of projectively presentable modules 18 5. Annihilator ideals of simple Whittaker modules 23 6. Linkage principle of the category N Z for gl(m|n), osp(2|2n) and pe(n) 32 References 40

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“…Proof. We adapt the argument in [CM,Lemma 4.2] to complete the proof. Since U (g) is a finitely-generated U (g0)-module, the g0-module Res S is finitely-generated.…”

Section: 2mentioning

confidence: 99%

Whittaker modules for classical Lie superalgebras

Chen

2021

Preprint

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We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić-Soergel equivalence of a category of Whittaker modules and a category of Harish-Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories O. As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras gl(m|n) and the ortho-symplectic Lie superalgebras osp(2|2n) can be computed via the Kazhdan-Lusztig combinatorics.

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Simple supermodules over Lie superalgebras

Cited by 20 publications

References 27 publications

Classification of simple Harish-Chandra modules for basic classical map superalgebras

Classification of simple Harish-Chandra modules for basic classical map superalgebras

Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras

Whittaker modules for classical Lie superalgebras

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