Quasi-Whittaker modules over the conformal Galilei algebras

Cai, Yong; Chen, Qiufan

doi:10.1080/03081087.2016.1183561

Cited by 1 publication

(3 citation statements)

References 18 publications

(22 reference statements)

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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%

“…Several papers studied a generalization of Whittaker modules (originally defined in [37] for semisimple Lie algebras), in the setup of conformal Galilei algebras and the Schrödinger Lie algebra (see [10]- [12], [40]). Quasi-Whittaker modules are modules on which the radical of the Lie algebra acts locally finitely.…”

Section: §1 Introduction and Description Of The Resultsmentioning

confidence: 99%

“…By adjunction, this must have both V (2, λ) and V (2, λ) as simple subquotients, both with multiplicity 1. The remaining g-types are L(4), L(8), L (10). .…”

Section: Proof Let L(λ) → Mmentioning

confidence: 99%

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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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