Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915013216698368 |
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| author | Liu, Genqiang Li, Mingjie |
| author_facet | Liu, Genqiang Li, Mingjie |
| contents | Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{ε_1-ε_2},\dots,X_{ε_1-ε_n}, X_{2ε_1}$. We show that the minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n}, e)$ is isomorphic to the centralizer $C_{U_S}(B)$ of some subalgebra $B$ in $U_S$, and it can be identified with a tensor product factor of $U_S$. As an application, we show that the category of weight $\mathfrak{sp}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W(\mathfrak{sp}_{2n}, e)$, explaining the coincidence that both of them are semi-simple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$ Liu, Genqiang Li, Mingjie Representation Theory Rings and Algebras Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{ε_1-ε_2},\dots,X_{ε_1-ε_n}, X_{2ε_1}$. We show that the minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n}, e)$ is isomorphic to the centralizer $C_{U_S}(B)$ of some subalgebra $B$ in $U_S$, and it can be identified with a tensor product factor of $U_S$. As an application, we show that the category of weight $\mathfrak{sp}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W(\mathfrak{sp}_{2n}, e)$, explaining the coincidence that both of them are semi-simple. |
| title | Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$ |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2411.06768 |