Lie algebras MOC

Normalizer in a Lie algebra

Let 𝔤 be a Lie algebra over 𝕂 and 𝑉 ≤𝔤 be a vector subspace. The normalizer 𝔫𝔤(𝑉) of 𝑉 in 𝔤 is the Lie subalgebra of all elements whose adjoint representations leave 𝑉 invariant, lie i.e.

𝔫𝔤(𝑉)={𝑥∈𝑉:[𝑥,𝑉]≤𝑉}

Proof of Lie subalgebra

Let 𝑥,𝑦 ∈𝔫𝔤(𝑉). Then by the ^Jacobi,

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

as required.

A subalgebra is the Centralizer in a Lie algebra 𝔠𝔤(𝑉) ≤𝔫𝔤(𝑉).

Further terminology

• A subalgebra 𝔥 ≤𝔤 is called self-normalizing iff 𝔫𝔤(𝔥) =𝔥.

Properties

1. 𝔫𝔤(𝑉) =𝔤 iff 𝑉 is a Lie algebra ideal of 𝔤

See also

• Normalizer in a group

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