Centralizer in a Lie algebra

Let $𝔤$ be a Lie algebra over $𝕂$ and $𝑉 \leq 𝔤$ be a vector subspace. The centralizer $𝔠 𝔤 (𝑉)$ of $𝑉$ in $𝔤$ is then the Lie subalgebra of elements giving zero bracket with all elements of $𝑉$ , #m/def/lie i.e.

𝔠 𝔤 (𝑉) = {𝑥 \in 𝔤 : [𝑥, 𝑉] = 0}

Proof of Lie subalgebra

Let $𝑥, 𝑦 \in 𝔠 𝔤 (𝑉)$ . By the Jacobi identity

[[𝑥, 𝑦], 𝑉] = [𝑥, [𝑦, 𝑉]] + [𝑦, [𝑉, 𝑥]] = 0

whence $[𝑥, 𝑦] \in 𝔠 𝔤 (𝑉)$ .

A related notion is the Centre of a Lie algebra $𝔷 (𝔤) = 𝔠 𝔤 (𝔤)$ , which includes only those elements that give zero bracket for all elements. A weakening is the Normalizer in a Lie algebra.

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