We study the representation theory of finite-dimensional ω-Lie algebras over the complex field. We derive an ω-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble ω-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D ω-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra 𝔤 and prove that if 𝔤 is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of 𝔤 and 1D ω-extensions of 𝔤.

中文翻译：

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ω-李代数的表示和李代数的有尾导数

我们研究有限维的表示论ω- 复数域上的李代数。我们得出一个ω- 经典李定理的李版本，即可溶的任何有限维不可约模ω-李代数是一维的（1D）。我们还证明了一些 3D 的不可分解模块ω-李代数可以由复场和幂零矩阵参数化。我们引入了非结合代数的有尾推导的概念𝔤并证明如果𝔤是一个李代数，那么有尾导数之间存在一一对应关系𝔤和一维ω- 扩展𝔤.