Generalized derivations of Complex $ω$-Lie Superalgebras
Jia Zhou
TL;DR
The paper extends derivation theory to finite-dimensional complex ω-Lie superalgebras by defining and interrelating generalized derivations, quasiderivations, centroids, and quasicentroids, and proves that $GDer^ω(g) = QDer^ω(g) + QCent^ω(g)$. It shows that compatible quasiderivations embed as derivations in a larger ω-Lie superalgebra $reve{g}$, and, when ${ m Ann}(g)=0$, yields a direct-sum decomposition ${ m Der}^ω(reve{g}) = φ(QDer^ω(g)) \,⊕\, ZDer(reve{g})$. The authors also provide detailed computations in the 3-dimensional case for the unique nontrivial ω-Lie superalgebra $H$, determining the dimensions and Jordan forms of ${ m GDer}(H)$, ${ m GDer}^{ω}(H)$, ${ m QDer}(H)$ and ${ m QDer}^{ω}(H)$, and highlighting exceptional behavior of $H$ relative to other 3D ω-Lie algebras. Overall, the work advances the structural understanding of derivations in ω-Lie superalgebras and yields tools for studying extensions and representations.
Abstract
~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: ${\rm Der}^ω(\breve{g})=\varphi({\rm QDer}^ω(g))\oplus {\rm ZDer}(\breve{g})$, 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.