A new quasi-lisse affine vertex algebra of type $D_4$
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| Format: | Preprint |
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2025
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| _version_ | 1866913103140093952 |
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| author | Adamović, Dražen Vukorepa, Ivana |
| author_facet | Adamović, Dražen Vukorepa, Ivana |
| contents | We consider a family of potential quasi-lisse affine vertex algebras $L_{k_m}(D_4)$ at levels $k_m =-6 + \frac{4}{2m+1}$. In the case $m=0$, the irreducible $L_{k_0}(D_4)$--modules were classified in arXiv:1205.3003, and it was proved in arXiv:1610.05865 that $L_{k_0}(D_4)$ is a quasi-lisse vertex algebra. We conjecture that $L_{k_m}(D_4)$ is quasi-lisse for every $m \in {\mathbb{Z}}_{>0}$, and that it contains a unique irreducible ordinary module. In this article we prove this conjecture for $m=1$, by using mostly computational methods. We show that the maximal ideal in the universal affine vertex algebra $V^{k_1}(D_4)$ is generated by three singular vectors of conformal weight six. The explicit formulas were obtained using software. Then we apply Zhu's theory and classify all irreducible $L_{k_1}(D_4)$--modules. It turns out that $L_{k_1}(D_4)$ has $405$ irreducible modules in the category $\mathcal O$, but a unique irreducible ordinary module. Finally, we prove that $L_{k_1}(D_4)$ is quasi-lisse by showing that its associated variety is contained in the nilpotent cone of $D_4$. We also prove that the associated variety $X_{L_{k_1}(D_4)}$ is $\overline{\mathbb O}_{sreg}$, the Zariski closure of the subregular nilpotent orbit in $D_4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new quasi-lisse affine vertex algebra of type $D_4$ Adamović, Dražen Vukorepa, Ivana Quantum Algebra Mathematical Physics Representation Theory We consider a family of potential quasi-lisse affine vertex algebras $L_{k_m}(D_4)$ at levels $k_m =-6 + \frac{4}{2m+1}$. In the case $m=0$, the irreducible $L_{k_0}(D_4)$--modules were classified in arXiv:1205.3003, and it was proved in arXiv:1610.05865 that $L_{k_0}(D_4)$ is a quasi-lisse vertex algebra. We conjecture that $L_{k_m}(D_4)$ is quasi-lisse for every $m \in {\mathbb{Z}}_{>0}$, and that it contains a unique irreducible ordinary module. In this article we prove this conjecture for $m=1$, by using mostly computational methods. We show that the maximal ideal in the universal affine vertex algebra $V^{k_1}(D_4)$ is generated by three singular vectors of conformal weight six. The explicit formulas were obtained using software. Then we apply Zhu's theory and classify all irreducible $L_{k_1}(D_4)$--modules. It turns out that $L_{k_1}(D_4)$ has $405$ irreducible modules in the category $\mathcal O$, but a unique irreducible ordinary module. Finally, we prove that $L_{k_1}(D_4)$ is quasi-lisse by showing that its associated variety is contained in the nilpotent cone of $D_4$. We also prove that the associated variety $X_{L_{k_1}(D_4)}$ is $\overline{\mathbb O}_{sreg}$, the Zariski closure of the subregular nilpotent orbit in $D_4$. |
| title | A new quasi-lisse affine vertex algebra of type $D_4$ |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2504.13783 |