Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras
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| Format: | Preprint |
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2024
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| _version_ | 1866916461088342016 |
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| author | Chen, Chih-Whi Cheng, Shun-Jen Suh, Uhi Rinn |
| author_facet | Chen, Chih-Whi Cheng, Shun-Jen Suh, Uhi Rinn |
| contents | For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimesΛ(θ)$, where $θ$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $χ$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $χ$. In the case when $χ$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras Chen, Chih-Whi Cheng, Shun-Jen Suh, Uhi Rinn Representation Theory 17B10, 17B20 For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimesΛ(θ)$, where $θ$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $χ$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $χ$. In the case when $χ$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra. |
| title | Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras |
| topic | Representation Theory 17B10, 17B20 |
| url | https://arxiv.org/abs/2410.22819 |