Quasi-Whittaker modules
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916887118479360 |
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| author | Cheng, Cunguang Gao, Wenting Liu, Shiyuan Zhao, Kaiming Zhao, Yueqiang |
| author_facet | Cheng, Cunguang Gao, Wenting Liu, Shiyuan Zhao, Kaiming Zhao, Yueqiang |
| contents | In this paper, a general setting is proposed to define a class of modules over nonsemisimple Lie algebras $\mathfrak{g}$ induced by a nonperfect ideal $\mathfrak{p}$. This class of Lie algebras includes many well-known Lie algebras, and some of this class of modules are Whittaker modules and others are not. We call these modules quasi-Whittaker modules. By introducing a new concept: the Whittaker annihilator for universal quasi-Whittaker modules, we are able to determine the necessary and sufficient conditions for the irreducibility of the universal quasi-Whittaker modules. In the reducible case, we can obtain some maximal submodules. In particular, we classify the irreducible quasi-Whittaker modules for many Lie algebras, and obtain a lot of irreducible smooth $\mathcal{W}_n^+$-modules of height $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-Whittaker modules Cheng, Cunguang Gao, Wenting Liu, Shiyuan Zhao, Kaiming Zhao, Yueqiang Representation Theory Quantum Algebra Rings and Algebras 17B10, 17B20, 17B65 In this paper, a general setting is proposed to define a class of modules over nonsemisimple Lie algebras $\mathfrak{g}$ induced by a nonperfect ideal $\mathfrak{p}$. This class of Lie algebras includes many well-known Lie algebras, and some of this class of modules are Whittaker modules and others are not. We call these modules quasi-Whittaker modules. By introducing a new concept: the Whittaker annihilator for universal quasi-Whittaker modules, we are able to determine the necessary and sufficient conditions for the irreducibility of the universal quasi-Whittaker modules. In the reducible case, we can obtain some maximal submodules. In particular, we classify the irreducible quasi-Whittaker modules for many Lie algebras, and obtain a lot of irreducible smooth $\mathcal{W}_n^+$-modules of height $2$. |
| title | Quasi-Whittaker modules |
| topic | Representation Theory Quantum Algebra Rings and Algebras 17B10, 17B20, 17B65 |
| url | https://arxiv.org/abs/2508.05917 |