Representations of omega-Lie algebras and tailed derivations of Lie algebras

摘要

We study the representation theory of finite-dimensional omega-Lie algebras over the complex field. We derive an omega-Lie version of the classical Lie's theorem, i.e., any finite-dimensional irreducible module of a soluble omega-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D omega-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 omega-extensions of g.