Lie algebras MOC

Normalizer in a Lie algebra

Let $𝔤$ be a Lie algebra over $𝕂$ and $𝑉 \leq 𝔤$ be a vector subspace. The normalizer $𝔫 𝔤 (𝑉)$ of $𝑉$ in $𝔤$ is the Lie subalgebra of all elements whose adjoint representations leave $𝑉$ invariant, #m/def/lie i.e.

𝔫 𝔤 (𝑉) = {𝑥 \in 𝑉 : [𝑥, 𝑉] \leq 𝑉}

Proof of Lie subalgebra

Let $𝑥, 𝑦 \in 𝔫 𝔤 (𝑉)$ . Then by the Jacobi identity,

[[𝑥, 𝑦], 𝑉] = [𝑥, [𝑦, 𝑉]] + [𝑦, [𝑉, 𝑥]] \subseteq [𝑥, 𝑉] + [𝑦, 𝑉] \subseteq 𝑉

as required.

A subalgebra is the Centralizer in a Lie algebra $𝔠 𝔤 (𝑉) \leq 𝔫 𝔤 (𝑉)$ .

Further terminology

A subalgebra $𝔥 \leq 𝔤$ is called self-normalizing iff $𝔫 𝔤 (𝔥) = 𝔥$ .

Properties

$𝔫 𝔤 (𝑉) = 𝔤$ iff $𝑉$ is a Lie algebra ideal of $𝔤$

See also

Normalizer in a group

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