On the transitivity of Lie ideals and a characterization of 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 $\mathfrak{g}$.