Quasi-lisse extension of affine $\mathfrak{sl}_2$ à la Feigin--Tipunin
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910458360889344 |
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| author | Creutzig, Thomas Nakatsuka, Shigenori Sugimoto, Shoma |
| author_facet | Creutzig, Thomas Nakatsuka, Shigenori Sugimoto, Shoma |
| contents | We study the affine analogue $\mathrm{FT}_p(\mathfrak{sl}_2)$ of the triplet algebra. We show that $\mathrm{FT}_p(\mathfrak{sl}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of $\mathfrak{sl}_2$. We realize $\mathrm{FT}_p(\mathfrak{sl}_2)$ as the global sections of a sheaf of vertex algebras in the spirit of Feigin--Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama--Suzuki dual superalgebra $s\mathcal{W}_p(\mathfrak{sl}_{2|1})$ of $\mathrm{FT}_p(\mathfrak{sl}_2)$ and their singlet type subalgebras $s\mathcal{M}_p(\mathfrak{sl}_{2|1})$ and $\mathcal{M}_p(\mathfrak{sl}_2)$ and show their correspondence of categories. For $p=1$, we show the logarithmic Kazhdan--Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to $s\mathcal{M}_1(\mathfrak{sl}_{2|1})$ is the unrolled restricted quantum supergroup $u^H_{-1}(\mathfrak{sl}_{2|1})$ as suggested by Semikhatov and Tipunin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_13568 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasi-lisse extension of affine $\mathfrak{sl}_2$ à la Feigin--Tipunin Creutzig, Thomas Nakatsuka, Shigenori Sugimoto, Shoma Representation Theory Mathematical Physics Quantum Algebra We study the affine analogue $\mathrm{FT}_p(\mathfrak{sl}_2)$ of the triplet algebra. We show that $\mathrm{FT}_p(\mathfrak{sl}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of $\mathfrak{sl}_2$. We realize $\mathrm{FT}_p(\mathfrak{sl}_2)$ as the global sections of a sheaf of vertex algebras in the spirit of Feigin--Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama--Suzuki dual superalgebra $s\mathcal{W}_p(\mathfrak{sl}_{2|1})$ of $\mathrm{FT}_p(\mathfrak{sl}_2)$ and their singlet type subalgebras $s\mathcal{M}_p(\mathfrak{sl}_{2|1})$ and $\mathcal{M}_p(\mathfrak{sl}_2)$ and show their correspondence of categories. For $p=1$, we show the logarithmic Kazhdan--Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to $s\mathcal{M}_1(\mathfrak{sl}_{2|1})$ is the unrolled restricted quantum supergroup $u^H_{-1}(\mathfrak{sl}_{2|1})$ as suggested by Semikhatov and Tipunin. |
| title | Quasi-lisse extension of affine $\mathfrak{sl}_2$ à la Feigin--Tipunin |
| topic | Representation Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2306.13568 |