Generalized derivations of $ω$-Lie algebras
Yin Chen, Shan Ren, Jiawen Shan, Runxuan Zhang
TL;DR
This work develops a structural theory for generalized derivations and compatible variants in finite-dimensional $\omega$-Lie algebras, extending the classical tower of derivation spaces to $\mathrm{Der}_c(L)\subseteq\mathrm{GDer}_c(L)\subseteq\mathrm{GDer}(L)\subseteq\mathfrak{gl}(L)$ and linking it to automorphism-type questions. It proves that any compatible quasiderivation of an $\omega$-Lie algebra $L$ can be embedded as a compatible derivation into a larger $\omega$-Lie algebra $\widetilde{L}$ via an injective Lie-homomorphism $\updelta_U:\mathrm{QDer}(L)\to\mathrm{Der}(\widetilde{L})$, and establishes a decomposition $\mathrm{Der}_c(\widetilde{L})=\updelta_U(\mathrm{QDer}_c(L))\oplus\mathrm{ZDer}(\widetilde{L})$ (centered on the condition $c(L)=0$). The paper also provides an explicit, computation-driven framework to determine $\mathrm{GDer}(L)$, $\mathrm{GDer}_c(L)$, $\mathrm{QDer}(L)$, and $\mathrm{QDer}_c(L)$ for all 3-dimensional non-Lie complex $\omega$-Lie algebras, exploiting the CLZ14 classification; in these cases $\mathrm{QDer}(L)=\mathrm{GDer}(L)$ and $\mathrm{QDer}_c(L)=\mathrm{GDer}_c(L)$. Collectively, these results extend LL00’s embedding paradigm to the $\omega$-Lie setting and furnish useful computational tools for low-dimensional $\omega$-Lie algebras.
Abstract
This article explores the structure theory of compatible generalized derivations of finite-dimensional $ω$-Lie algebras over a field $\mathbb{K}$. We prove that any compatible quasiderivation of an $ω$-Lie algebra can be embedded as a compatible derivation into a larger $ω$-Lie algebra, refining the general result established by Leger and Luks in 2000 for finite-dimensional nonassociative algebras. We also provide an approach to explicitly compute (compatible) generalized derivations and quasiderivations for all $3$-dimensional non-Lie complex $ω$-Lie algebras.