Representations of $\omega$-Lie Algebras and Tailed Derivations of Lie Algebras

TL;DR

This paper explores the representation theory of finite-dimensional $\omega$-Lie algebras, establishing an $\omega$-Lie version of Lie's theorem and linking tailed derivations to $\omega$-extensions of Lie algebras.

Contribution

It introduces the $\omega$-Lie version of Lie's theorem, classifies indecomposable modules for certain $\omega$-Lie algebras, and defines tailed derivations with a correspondence to $\omega$-extensions.

Findings

Finite-dimensional irreducible modules of soluble $\omega$-Lie algebras are one-dimensional.

Indecomposable modules of some three-dimensional $\omega$-Lie algebras are parametrized by complex numbers and nilpotent matrices.

Tailed derivations of a Lie algebra correspond to one-dimensional $\omega$-extensions.

Abstract

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 one-dimensional. We also prove that indecomposable modules of some three-dimensional $\omega$-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 one-dimensional $\omega$-extensions of $g$.