Ideal of a Lie algebra

This article defines a property for a subalgebra of a Lie algebra

ANALOGY: This is an analogue in Lie algebra of a property encountered in group. Specifically, it is a Lie subalgebra property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in Lie algebras of subgroup properties (OR, View as a tabulated list)

Further information: Ideal of a Lie ring

Definition

Definition with symbols

A subset $B$ of a Lie algebra $A$ is said to be an ideal if it is a vector subspace of $A$ under addition, and $[x,y] \in B$ for any $x \in B$ and $y \in A$. Note that any ideal is, in particular, also a Lie subalgebra.

Relation with the Lie group

The Lie algebra of any closed normal subgroup of a Lie group, is an ideal. This follows from the following:

The commutator of $v$ and $w$ is the differential at $t = 0$ of $\exp(tv).w - w$. Here $\exp(tv)$ acts on $w$ via the adjoint representation.

Now if $w$ is an element in the Lie algebra of a closed normal subgroup $N$, then $\exp(tv)\exp(t'w)\exp(tv)^{-1}$ is also in $N$ for every $t'$, and differentiating at $t'=0$ gives $\exp{tv}.w$ to be in the Lie algebra of $N$. Thus, the difference $\exp(tv).w - w$ is also in the Lie algebra, and hence so is the value $[v,w]$

Thus $w$ being in the Lie algebra of $N$ implies that $[v,w]$ is. By the amticommutativity of the Lie bracket, we conclude that $v$ being in the Lie algebra implies $[v,w]$ is.