Ideal in a Lie algebra - Imported Knowlpedia Content

Let $\mathfrak g$ be a Lie algebra.

Definition (Ideal). A linear subspace $\mathfrak i\subseteq \mathfrak g$ is an ideal if it is a Lie subalgebra and

[\mathfrak g,\mathfrak i]\subseteq \mathfrak i.

Equivalently, $\mathfrak i$ is stable under the adjoint action: for every $x\in\mathfrak g$ , the endomorphism $\mathrm{ad}_x$ maps $\mathfrak i$ into itself.

Remarks

Basic consequences.

If $\phi:\mathfrak g\to\mathfrak h$ is a Lie algebra homomorphism, then $\ker\phi$ is an ideal.
If $\mathfrak i$ is an ideal, the quotient vector space $\mathfrak g/\mathfrak i$ carries a natural quotient Lie algebra structure.

Examples. The center $Z(\mathfrak g)$ is an ideal, and the derived subalgebra $[\mathfrak g,\mathfrak g]$ is an ideal (see the ideal lemma).

Context. Ideals play the role of normal subgroups in Lie theory and are central in structure results such as the Levi decomposition.