Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras
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| Format: | Preprint |
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2025
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| _version_ | 1866909937581424640 |
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| author | Lin, Jie Liu, Chengyu Jiang, Jingjing |
| author_facet | Lin, Jie Liu, Chengyu Jiang, Jingjing |
| contents | We investigate the transposed Poisson structures on both the $q$-analog Virasoro-like algebra and $q$-quantum torus Lie algebra considering the cases where $q$ is generic and where $q$ is a primitive root of unity, respectively. We establish the following results: When $q$ is generic, there are no non-trivial $\frac{1}{2}$-derivations and consequently, no non-trivial transposed Poisson algebra structures exist on the $q$-analog Virasoro-like algebra. Meanwhile, the $q$-quantum torus Lie algebra does possess non-trivial $\frac{1}{2}$-derivations but lacks of a non-trivial transposed Poisson structure. When $q$ is a primitive root of unity, both the $q$-analog Virasoro-like algebra and the $q$-quantum torus Lie algebra possess non-trivial $\frac{1}{2}$-derivations. We present the non-trivial transposed Poisson algebra structure for the $q$-analog Virasoro-like algebra. However, the $q$-quantum torus Lie algebra lacks of a non-trivial transposed Poisson structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras Lin, Jie Liu, Chengyu Jiang, Jingjing Rings and Algebras We investigate the transposed Poisson structures on both the $q$-analog Virasoro-like algebra and $q$-quantum torus Lie algebra considering the cases where $q$ is generic and where $q$ is a primitive root of unity, respectively. We establish the following results: When $q$ is generic, there are no non-trivial $\frac{1}{2}$-derivations and consequently, no non-trivial transposed Poisson algebra structures exist on the $q$-analog Virasoro-like algebra. Meanwhile, the $q$-quantum torus Lie algebra does possess non-trivial $\frac{1}{2}$-derivations but lacks of a non-trivial transposed Poisson structure. When $q$ is a primitive root of unity, both the $q$-analog Virasoro-like algebra and the $q$-quantum torus Lie algebra possess non-trivial $\frac{1}{2}$-derivations. We present the non-trivial transposed Poisson algebra structure for the $q$-analog Virasoro-like algebra. However, the $q$-quantum torus Lie algebra lacks of a non-trivial transposed Poisson structure. |
| title | Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2512.01299 |