Structure of sympathetic 3-Lie algebras

Abstract

In this paper, we will introduce the concept of sympathetic [Formula: see text]-Lie algebras and show that some classical properties of semi-simple [Formula: see text]-Lie algebras are still valid for sympathetic [Formula: see text]-Lie algebras. We prove that every perfect [Formula: see text]-Lie algebra [Formula: see text] contains a greatest special sympathetic ideal [Formula: see text], and there exists a solvable ideal of [Formula: see text] denoted by [Formula: see text] which is the greatest among the solvable ideals [Formula: see text] of [Formula: see text] for which [Formula: see text]. And we show that there exists a sympathetic subalgebra [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a sympathetic [Formula: see text]-Lie algebra if and only if [Formula: see text]. Moreover, we also study the ideals [Formula: see text] of a [Formula: see text]-Lie algebra [Formula: see text] such that [Formula: see text] is a sympathetic [Formula: see text]-Lie algebra and investigate some properties about them.