Typical representations of Takiff superalgebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929634388475904 |
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| author | Chen, Chih-Whi Wang, Yongjie |
| author_facet | Chen, Chih-Whi Wang, Yongjie |
| contents | We investigate representations of the $\ell$-th Takiff superalgebras $\widetilde{\mathfrak g}_\ell := \widetilde{\mathfrak g}\otimes \mathbb C[θ]/(θ^{\ell+1})$, for $\ell>0$, associated with a basic classical and a periplectic Lie superalgebras $\widetilde{\mathfrak g}$. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras $\widetilde{\mathfrak g}_\ell$. As a consequence, we provide a complete description of the characters of the finite-dimensional modules over type I Takiff superalgebras. For the Lie superalgebras $\widetilde{\mathfrak g}= \mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, we prove that the Kac induction functor of $\widetilde{\mathfrak g}_\ell$ leads to an equivalence from an arbitrary typical Jordan block of the category $\mathcal O$ for $\widetilde{\mathfrak g}_\ell$ to a Jordan block of the category $\mathcal O$ for the even subalgebra of $\widetilde{\mathfrak g}_\ell$. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07894 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Typical representations of Takiff superalgebras Chen, Chih-Whi Wang, Yongjie Representation Theory 17B10, 17B55 We investigate representations of the $\ell$-th Takiff superalgebras $\widetilde{\mathfrak g}_\ell := \widetilde{\mathfrak g}\otimes \mathbb C[θ]/(θ^{\ell+1})$, for $\ell>0$, associated with a basic classical and a periplectic Lie superalgebras $\widetilde{\mathfrak g}$. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras $\widetilde{\mathfrak g}_\ell$. As a consequence, we provide a complete description of the characters of the finite-dimensional modules over type I Takiff superalgebras. For the Lie superalgebras $\widetilde{\mathfrak g}= \mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, we prove that the Kac induction functor of $\widetilde{\mathfrak g}_\ell$ leads to an equivalence from an arbitrary typical Jordan block of the category $\mathcal O$ for $\widetilde{\mathfrak g}_\ell$ to a Jordan block of the category $\mathcal O$ for the even subalgebra of $\widetilde{\mathfrak g}_\ell$. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras. |
| title | Typical representations of Takiff superalgebras |
| topic | Representation Theory 17B10, 17B55 |
| url | https://arxiv.org/abs/2404.07894 |