Generalized derivations of Complex $ω$-Lie Superalgebras

Abstract: ~Let $(g,~[-,-],~\omega)$ be a finite-dimensional complex $\omega$-Lie superalgebra. This paper explores the algbaraic structures of generalized derivation superalgebra ${\rm GDer}(g)$, compatatible generalized derivations algebra ${\rm GDer}^{\omega}(g)$, and their subvarieties such as quasiderivation superalgebra ${\rm QDer}(g)$(${\rm QDer}^{\omega}(g)$), centroid ${\rm Cent}(g)$ (${\rm Cent}^{\omega}(g)$) and quasicentroid ${\rm QCent}(g)$ (${\rm QCent}^{\omega}(g)$). We prove that ${\rm GDer}^{\omega}(g) = {\rm QDer}^{\omega}(g) + {\rm QCent}^{\omega}(g)$. We also study the embedding question of compatible quasiderivations of $\omega$-Lie superalgebras, demonstrating that ${\rm QDer}^{\omega}(g)$ can be embedded as derivations in a larger $\omega$-Lie superalgebra $\breve g$ and furthermore, we obtain a semidirect sum decomposition: ${\rm Der}^{\omega}(\breve{g})=\varphi({\rm QDer}^{\omega}(g))\oplus {\rm ZDer}(\breve{g})$, when the annihilator of $g$ is zero. In particular, for the 3-dimensional complex $\omega$-Lie superalgebra $H$, we explicitly calculate ${\rm GDer}(H)$, ${\rm GDer}^{\omega}(H)$, ${\rm QDer}(H)$ and ${\rm QDer}^{\omega}(H)$, and derive the Jordan standard forms of generic elements in these varieties.