Ideal of a Lie algebra

An ideal of a Lie algebra $\mathfrak{g}$ is a subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ such that

Equivalently, $[\mathfrak{g}, \mathfrak{h}] \subseteq \mathfrak{h}$.

Properties

• Every ideal is a subalgebra (but not conversely).
• The quotient $\mathfrak{g}/\mathfrak{h}$ inherits a Lie algebra structure when $\mathfrak{h}$ is an ideal.
• The kernel of any Lie algebra homomorphism is an ideal.

Examples

• The center $Z(\mathfrak{g}) = \{X : [X, Y] = 0 \text{ for all } Y\}$.
• The derived algebra $[\mathfrak{g}, \mathfrak{g}]$.
• $\{0\}$ and $\mathfrak{g}$ itself (trivial ideals).

Simple and semisimple

• A Lie algebra is simple if it has no proper nonzero ideals and $\dim \mathfrak{g} > 1$.
• A Lie algebra is semisimple if it has no nonzero abelian ideals.