Derived subalgebra

The Lie subalgebra spanned by commutators; it measures how far is from abelian.

Derived subalgebra

Let $\mathfrak g$ be a Lie algebra .

Definition

The derived subalgebra (or commutator subalgebra) of $\mathfrak g$ is

[\mathfrak g,\mathfrak g] := \mathrm{span}\{[x,y] : x,y\in \mathfrak g\}\subseteq \mathfrak g.

It is a Lie subalgebra, and in fact an ideal ; the ideal property is recorded explicitly in the lemma that $[\mathfrak g,\mathfrak g]$ is an ideal .

Basic consequences

$\mathfrak g$ is abelian iff $[\mathfrak g,\mathfrak g]=0$ .
The quotient $\mathfrak g/[\mathfrak g,\mathfrak g]$ is the abelianization of $\mathfrak g$ (a special case of quotient Lie algebra ).
$\mathfrak g$ is called perfect if $[\mathfrak g,\mathfrak g]=\mathfrak g$ ; for example, any simple Lie algebra is perfect.

Context

The derived subalgebra is the first step in the derived series , which detects solvability and organizes many structure theorems such as the Levi decomposition .