The $\omega$-Lie algebra defined by the commutator of an $\omega$-left-symmetric algebra is not perfect

TL;DR

This paper investigates the relationship between $ ext{omega}$-Lie algebras and $ ext{omega}$-left-symmetric algebras, proving that perfect $ ext{omega}$-Lie algebras cannot be derived from the commutator of an $ ext{omega}$-left-symmetric algebra over the complex numbers.

Contribution

It establishes a non-existence result linking perfect $ ext{omega}$-Lie algebras to $ ext{omega}$-left-symmetric algebra structures, expanding understanding of their algebraic properties.

Findings

Perfect $ ext{omega}$-Lie algebras cannot be realized as commutators of $ ext{omega}$-left-symmetric algebras.

Classification of $ ext{omega}$-Lie algebras used to derive the main result.

Analysis over the complex numbers field $ ext{C}$.

Abstract

In this paper, we study admissible $\omega$-left-symmetric algebraic structures on $\omega$-Lie algebras over the complex numbers field $\mathbb C$. Based on the classification of $\omega$-Lie algebras, we prove that any perfect $\omega$-Lie algebra can't be the $\omega$-Lie algebra defined by the commutator of an $\omega$-left-symmetric algebra.