Quasi-derivations of Witt and related algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914014982832128 |
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| author | Kaygorodov, Ivan Khudoyberdiyev, Abror Shermatova, Zarina |
| author_facet | Kaygorodov, Ivan Khudoyberdiyev, Abror Shermatova, Zarina |
| contents | In the present work, we compute quasi-derivations of the Witt algebra and some algebras well-related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $\frac{1}{2}$-derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras ${\mathcal W}(a,b):$ in the case of $b=-1,$ they do not have interesting examples of quasi-derivations, but the case of $b\neq-1$ provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of ${\mathcal W}(a,b).$ As a corollary, we describe the derivations and quasi-derivations of the Novikov-Witt and admissible Novikov-Witt algebras previously constructed by Bai and his co-authors; and $δ$-derivations and transposed $δ$-Poisson structures on cited Lie algebras. In particular, we proved that each ${\mathcal W}(a,b)$ admits a nontrivial transposed $\frac 1{1-b}$-Poisson structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-derivations of Witt and related algebras Kaygorodov, Ivan Khudoyberdiyev, Abror Shermatova, Zarina Rings and Algebras In the present work, we compute quasi-derivations of the Witt algebra and some algebras well-related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $\frac{1}{2}$-derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras ${\mathcal W}(a,b):$ in the case of $b=-1,$ they do not have interesting examples of quasi-derivations, but the case of $b\neq-1$ provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of ${\mathcal W}(a,b).$ As a corollary, we describe the derivations and quasi-derivations of the Novikov-Witt and admissible Novikov-Witt algebras previously constructed by Bai and his co-authors; and $δ$-derivations and transposed $δ$-Poisson structures on cited Lie algebras. In particular, we proved that each ${\mathcal W}(a,b)$ admits a nontrivial transposed $\frac 1{1-b}$-Poisson structure. |
| title | Quasi-derivations of Witt and related algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2508.14914 |