Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Bavula, V. V.; Lü, Tao

doi:10.1063/1.4973378

Cited by 8 publications

(15 citation statements)

References 26 publications

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“…In contrast to this situation, for nonsemisimple Lie algebras, very little is known. Apart from the main result of [8], which, in addition to , classifies simple modules over the Borel subalgebra of , several special classes of simple modules were studied for various specific nonsemisimple Lie algebras (see, e.g., [6], [7], [10], [11], [20], [39], [43], [52] and references therein). We now look at some of these and some other results in more detail.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Lie Algebra Modules Which Are Locally Finite and With Finite Multiplicities Over the Semisimple Part

Mazorchuk¹,

Mrđen²

2021

Nagoya Math. J.

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For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

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Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Lie Algebra Modules Which Are Locally Finite and With Finite Multiplicities Over the Semisimple Part

Mazorchuk¹,

Mrđen²

2021

Nagoya Math. J.

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“…Recently, classifications of simple weight modules are obtained for some classical algebras (the Euclidean algebra, the Schrödinger algebra, the universal enveloping algebra U (sl 2 V 2 )), see [5][6][7]. In these classifications, classifications of all simple modules over certain subalgebras of the Weyl algebra A 1 that contain the polynomial algebra K [X ] (the, so-called, polyonic algebras) play a crucial role.…”

Section: V Bavula (B)mentioning

confidence: 99%

Classification of Simple Modules of the Ore Extension $$K[X][Y; f\frac{d}{dX}]$$

Bavula

2019

Math.Comput.Sci.

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“…The classification of simple weight modules over finite dimensiona semisimple Lie algebras with finite-dimensional weight spaces was obtained in 2000, see [M2]. Besides sl 2 (and its some deformations), all simple weight modules were constructed only for the aging algebra [LMZ], the Schrodinger algebra [BL2], and the Euclidean algebra [BL1].…”

Section: Introductionmentioning

confidence: 99%

Simple Witt Modules that are Finitely Generated over the Cartan Subalgebra

Guo¹,

Liu²,

Lü³

et al. 2020

MMJ

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Let d ≥ 1 be an integer, W d and K d be the Witt algebra and the weyl algebra over the Laurent polynomial algebra, respectively. For any gl d -module M and any admissible module P over the extended Witt algebra W d , we define a W d -module structure on the tensor product P ⊗ M . We prove in this paper that any simple W d -module that is finitely generated over the cartan subalgebra is a quotient module of the W d -module P ⊗ M for a finite dimensional simple gl d -module M and a simple K d -module P that are finitely generated over the cartan subalgebra. We also characterize all simple K d -modules and all simple admissible W d -modules that are finitely generated over the cartan subalgebra.

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Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Cited by 8 publications

References 26 publications

Lie Algebra Modules Which Are Locally Finite and With Finite Multiplicities Over the Semisimple Part

Lie Algebra Modules Which Are Locally Finite and With Finite Multiplicities Over the Semisimple Part

Classification of Simple Modules of the Ore Extension $$K[X][Y; f\frac{d}{dX}]$$

Simple Witt Modules that are Finitely Generated over the Cartan Subalgebra

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