$(d,σ)$-twisted Affine-Virasoro superalgebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911030348611584 |
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| author | Lü, Rencai You, Xizhou Zhao, Kaiming |
| author_facet | Lü, Rencai You, Xizhou Zhao, Kaiming |
| contents | For any finite dimensional Lie superalgebra $\dot{\mathfrak{g}}$ (maybe a Lie algebra) with an even derivation $d$ and a finite order automorphism $σ$ that commutes with $d$, we introduce the $(d,σ)$-twisted Affine-Virasoro superalgebra $\mathfrak{L}=\mathfrak{L}(\dot{\mathfrak{g}},d,σ)$ and determine its universal central extension $\hat{\mathfrak{L}}=\hat{\mathfrak{L}}(\dot{\mathfrak{g}},d,σ)$. This is a huge class of infinite-dimensional Lie superalgebras. Such Lie superalgebras consist of many new and well-known Lie algebras and superalgebras, including the Affine-Virasoro superalgebras, the twisted Heisenberg-Virasoro algebra, the mirror Heisenberg-Virasoro algebra, the W-algebra $W(2,2)$, the gap-$p$ Virasoro algebras, the Fermion-Virasoro algebra, the $N=1$ BMS superalgebra, the planar Galilean conformal algebra. Then we give the classification of cuspidal $A\mathfrak{L}$-modules by using the weighting functor from $U(\mathfrak{h})$-free modules to weight modules. Consequently, we give the classification of simple cuspidal $\mathfrak{L}$-modules by using the $A$-cover method. Finally, all simple quasi-finite modules over $\mathfrak{L}$ and $\hat{\mathfrak{L}}$ are classified. Our results recover many known Lie superalgebra results from mathematics and mathematical physics, and give many new Lie superalgebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(d,σ)$-twisted Affine-Virasoro superalgebras Lü, Rencai You, Xizhou Zhao, Kaiming Representation Theory Quantum Algebra Rings and Algebras 17B10, 17B20, 17B65, 17B66, 17B68 For any finite dimensional Lie superalgebra $\dot{\mathfrak{g}}$ (maybe a Lie algebra) with an even derivation $d$ and a finite order automorphism $σ$ that commutes with $d$, we introduce the $(d,σ)$-twisted Affine-Virasoro superalgebra $\mathfrak{L}=\mathfrak{L}(\dot{\mathfrak{g}},d,σ)$ and determine its universal central extension $\hat{\mathfrak{L}}=\hat{\mathfrak{L}}(\dot{\mathfrak{g}},d,σ)$. This is a huge class of infinite-dimensional Lie superalgebras. Such Lie superalgebras consist of many new and well-known Lie algebras and superalgebras, including the Affine-Virasoro superalgebras, the twisted Heisenberg-Virasoro algebra, the mirror Heisenberg-Virasoro algebra, the W-algebra $W(2,2)$, the gap-$p$ Virasoro algebras, the Fermion-Virasoro algebra, the $N=1$ BMS superalgebra, the planar Galilean conformal algebra. Then we give the classification of cuspidal $A\mathfrak{L}$-modules by using the weighting functor from $U(\mathfrak{h})$-free modules to weight modules. Consequently, we give the classification of simple cuspidal $\mathfrak{L}$-modules by using the $A$-cover method. Finally, all simple quasi-finite modules over $\mathfrak{L}$ and $\hat{\mathfrak{L}}$ are classified. Our results recover many known Lie superalgebra results from mathematics and mathematical physics, and give many new Lie superalgebras. |
| title | $(d,σ)$-twisted Affine-Virasoro superalgebras |
| topic | Representation Theory Quantum Algebra Rings and Algebras 17B10, 17B20, 17B65, 17B66, 17B68 |
| url | https://arxiv.org/abs/2507.00349 |