On 3-Lie algebras with abelian ideals and subalgebras

TL;DR

This paper investigates the structure and classification of 3-Lie algebras over algebraically closed fields, focusing on abelian subalgebras, ideals, and their dimensions, revealing new distinctions and classifications based on these properties.

Contribution

It provides new classifications of 3-Lie algebras based on the dimensions of abelian subalgebras and ideals, including cases with specific derived algebra dimensions.

Findings

Dimensions of abelian subalgebras and ideals do not always coincide in characteristic zero.

3-Lie algebras with maximal abelian ideals of dimension m-2 are 2-step solvable.

Complete classifications are given for 3-Lie algebras with certain maximal abelian subalgebra dimensions.

Abstract

In this paper, we study the maximal dimension $\alpha(L)$ of abelian subalgebras and the maximal dimension $\beta(L)$ of abelian ideals of m-dimensional 3-Lie algebras $L$ over an algebraically closed field. We show that these dimensions do not coincide if the field is of characteristic zero, even for nilpotent 3-Lie algebras. We then prove that 3-Lie algebras with $\beta(L) = m-2$ are 2-step solvable (see definition in Section 2). Furthermore, we give a precise description of these 3-Lie algebras with one or two dimensional derived algebras. In addition, we provide a classification of 3-Lie algebras with $\alpha(L)=\dim L-2$. We also obtain the classification of 3-Lie algebras with $\alpha(L)=\dim L-1$ and with their derived algebras of one dimension.