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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2505.22966 |
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| _version_ | 1866913865176973312 |
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| author | Zhou, Jia |
| author_facet | Zhou, Jia |
| contents | ~Let $(g,~[-,-],~ω)$ be a finite-dimensional complex $ω$-Lie superalgebra. This paper explores the algbaraic structures of generalized derivation superalgebra ${\rm GDer}(g)$, compatatible generalized derivations algebra ${\rm GDer}^ω(g)$, and their subvarieties such as quasiderivation superalgebra ${\rm QDer}(g)$(${\rm QDer}^ω(g)$), centroid ${\rm Cent}(g)$ (${\rm Cent}^ω(g)$) and quasicentroid ${\rm QCent}(g)$ (${\rm QCent}^ω(g)$). We prove that ${\rm GDer}^ω(g) = {\rm QDer}^ω(g) + {\rm QCent}^ω(g)$. We also study the embedding question of compatible quasiderivations of $ω$-Lie superalgebras, demonstrating that ${\rm QDer}^ω(g)$ can be embedded as derivations in a larger $ω$-Lie superalgebra $\breve g$ and furthermore, we obtain a semidirect sum decomposition: ${\rm Der}^ω(\breve{g})=φ({\rm QDer}^ω(g))\oplus {\rm ZDer}(\breve{g})$, when the annihilator of $g$ is zero. In particular, for the 3-dimensional complex $ω$-Lie superalgebra $H$, we explicitly calculate ${\rm GDer}(H)$, ${\rm GDer}^ω(H)$, ${\rm QDer}(H)$ and ${\rm QDer}^ω(H)$, and derive the Jordan standard forms of generic elements in these varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized derivations of Complex $ω$-Lie Superalgebras Zhou, Jia Rings and Algebras ~Let $(g,~[-,-],~ω)$ be a finite-dimensional complex $ω$-Lie superalgebra. This paper explores the algbaraic structures of generalized derivation superalgebra ${\rm GDer}(g)$, compatatible generalized derivations algebra ${\rm GDer}^ω(g)$, and their subvarieties such as quasiderivation superalgebra ${\rm QDer}(g)$(${\rm QDer}^ω(g)$), centroid ${\rm Cent}(g)$ (${\rm Cent}^ω(g)$) and quasicentroid ${\rm QCent}(g)$ (${\rm QCent}^ω(g)$). We prove that ${\rm GDer}^ω(g) = {\rm QDer}^ω(g) + {\rm QCent}^ω(g)$. We also study the embedding question of compatible quasiderivations of $ω$-Lie superalgebras, demonstrating that ${\rm QDer}^ω(g)$ can be embedded as derivations in a larger $ω$-Lie superalgebra $\breve g$ and furthermore, we obtain a semidirect sum decomposition: ${\rm Der}^ω(\breve{g})=φ({\rm QDer}^ω(g))\oplus {\rm ZDer}(\breve{g})$, when the annihilator of $g$ is zero. In particular, for the 3-dimensional complex $ω$-Lie superalgebra $H$, we explicitly calculate ${\rm GDer}(H)$, ${\rm GDer}^ω(H)$, ${\rm QDer}(H)$ and ${\rm QDer}^ω(H)$, and derive the Jordan standard forms of generic elements in these varieties. |
| title | Generalized derivations of Complex $ω$-Lie Superalgebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2505.22966 |