Lie algebras MOC

Centralizer in a Lie algebra

Let 𝔤 be a Lie algebra over 𝕂 and 𝑉 ≤𝔤 be a vector subspace. The centralizer 𝔠𝔤(𝑉) of 𝑉 in 𝔤 is then the Lie subalgebra of elements giving zero bracket with all elements of 𝑉, lie i.e.

𝔠𝔤(𝑉)={𝑥∈𝔤:[𝑥,𝑉]=0}

Proof of Lie subalgebra

Let 𝑥,𝑦 ∈𝔠𝔤(𝑉). By the ^Jacobi

[[𝑥,𝑦],𝑉]=[𝑥,[𝑦,𝑉]]+[𝑦,[𝑉,𝑥]]=0

whence [𝑥,𝑦] ∈𝔠𝔤(𝑉).

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

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