Lie algebra ideal

A Lie algebra ideal $𝔞 ⊴ 𝔤$ is just an algebra ideal of a Lie algebra. #m/def/lie Equivalently, $𝔞 ⊴ 𝔤$ is a submodule under the adjoint representation. Note that a Lie subalgebra is a (two-sided) ideal iff it is a left or right ideal, by the alternating property. A Lie algebra is simple iff it has no nontrivial ideals.

Properties

Let $𝔞, 𝔟 ⊴ 𝔤$ be ideals. Then

$𝔞 + 𝔟$ is an ideal
$𝔞 \cap 𝔟$ is an ideal
$[𝔞, 𝔟]$ is an ideal (see Commutator ideal)

Proof

^P1 and ^P2 follow immediately. Note that for any $𝑥 \in 𝔤$ , by ^P1

a d 𝑥 [𝔞, 𝔟] = [a d 𝑥 𝔞, 𝔟] + [𝔞, a d 𝑥 𝔟] = [𝔞, 𝔟]

so $[𝔞, 𝔟]$ is an ideal.

Special ideals

See also

Quotient Lie algebra

#state/tidy | #lang/en | #SemBr