Whittaker categories and the minimal nilpotent finite $W$-algebra for $\mathfrak{sl}_{n+1}$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909135821340672 |
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| author | Liu, Genqiang Li, Yang |
| author_facet | Liu, Genqiang Li, Yang |
| contents | For any $\mathbf{a}=(a_1,\dots,a_n)\in \mathbb{C}^n$, we introduce a Whittaker category $\mathcal{H}_{\mathbf{a}}$ whose objects are $\mathfrak{sl}_{n+1}$-modules $M$ such that $e_{0i}-a_i$ acts locally nilpotently on $M$ for all $i \in \{1,\dots,n\}$, and the subspace $\mathrm{wh}_{\mathbf{a}}(M)=\{v\in M \mid e_{0i} v=a_iv, \ i=1,\dots,n\}$ is finite dimensional. In this paper, we first give a tensor product decomposition $U_S=W\otimes B$ of the localization $U_S$ of $U(\mathfrak{sl}_{n+1})$ with respect to the Ore subset $S$ generated by $e_{01},\dots, e_{0n}$. We show that the associative algebra $W$ is isomorphic to the type $A_n$ finite $W$-algebra $W(e)$ defined by a minimal nilpotent element $e$ in $\mathfrak{sl}_{n+1}$. Then using $W$-modules as a bridge, we show that each block with a generalized central character of $\mathcal{H}_{\mathbf{1}}$ is equivalent to the corresponding block of the cuspidal category $\mathcal{C}$, which is completely characterized by Grantcharov and Serganova. As a consequence, each regular integral block of $\mathcal{H}_{\mathbf{1}}$ and the category of finite dimensional modules over $W(e) can be described by a well-studied quiver with certain quadratic relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08791 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Whittaker categories and the minimal nilpotent finite $W$-algebra for $\mathfrak{sl}_{n+1}$ Liu, Genqiang Li, Yang Representation Theory Rings and Algebras For any $\mathbf{a}=(a_1,\dots,a_n)\in \mathbb{C}^n$, we introduce a Whittaker category $\mathcal{H}_{\mathbf{a}}$ whose objects are $\mathfrak{sl}_{n+1}$-modules $M$ such that $e_{0i}-a_i$ acts locally nilpotently on $M$ for all $i \in \{1,\dots,n\}$, and the subspace $\mathrm{wh}_{\mathbf{a}}(M)=\{v\in M \mid e_{0i} v=a_iv, \ i=1,\dots,n\}$ is finite dimensional. In this paper, we first give a tensor product decomposition $U_S=W\otimes B$ of the localization $U_S$ of $U(\mathfrak{sl}_{n+1})$ with respect to the Ore subset $S$ generated by $e_{01},\dots, e_{0n}$. We show that the associative algebra $W$ is isomorphic to the type $A_n$ finite $W$-algebra $W(e)$ defined by a minimal nilpotent element $e$ in $\mathfrak{sl}_{n+1}$. Then using $W$-modules as a bridge, we show that each block with a generalized central character of $\mathcal{H}_{\mathbf{1}}$ is equivalent to the corresponding block of the cuspidal category $\mathcal{C}$, which is completely characterized by Grantcharov and Serganova. As a consequence, each regular integral block of $\mathcal{H}_{\mathbf{1}}$ and the category of finite dimensional modules over $W(e) can be described by a well-studied quiver with certain quadratic relations. |
| title | Whittaker categories and the minimal nilpotent finite $W$-algebra for $\mathfrak{sl}_{n+1}$ |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2304.08791 |