On the transitivity of Lie ideals and a characterization of perfect Lie algebras

TL;DR

This paper investigates conditions under which Lie subideals are actually ideals, and characterizes perfect Lie algebras through the transitivity of the Lie ideal relation, providing intrinsic criteria for their identification.

Contribution

It introduces intrinsic and extrinsic conditions for the transitivity of Lie ideals and characterizes perfect Lie algebras via this transitivity property.

Findings

Perfect Lie algebras are characterized by the transitivity of the Lie ideal relation.

Subideals of perfect Lie algebras are necessarily ideals.

The paper provides intrinsic criteria for identifying perfect Lie algebras.

Abstract

We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being a Lie ideal, in the sense that if a Lie algebra $\mathfrak{h}$ is a subideal of a Lie algebra $\mathfrak{g}$ (i.e. there exist Lie subalgebras $\mathfrak{l}_0,\mathfrak{l}_1,\dots,\mathfrak{l}_n$ of $\mathfrak{g}$ with $\mathfrak{h}=\mathfrak{l}_0\unlhd \mathfrak{l}_1 \unlhd\cdots \unlhd \mathfrak{l}_n=\mathfrak{g}$), then $\mathfrak{h}$ is an ideal of $\mathfrak{g}$. We also prove that perfect Lie algebras of arbitrary dimension and over any field are intrinsically characterized by transitivity of this type; In particular, we show that a Lie algebra $\mathfrak{h}$ is perfect (i.e. $\mathfrak{h}=[\mathfrak{h}, \mathfrak{h}]$) if and only if for any Lie algebra $\mathfrak{g}$ such that $\mathfrak{h}$ is a subideal of $\mathfrak{g}$, it follows that $\mathfrak{h}$ is an ideal of $\mathfrak{g}$.