Generalized derivations of Complex $\ddot{I}‰$-Lie Superalgebras

Abstract

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