Centre of a Lie algebra

The centre $𝔷 (𝔤)$ of a Lie algebra $𝔤$ is a Lie algebra ideal consisting of elements which annihilate all elements in the Lie bracket, #m/def/lie i.e.

𝔷 (𝔤) = {𝑥 \in 𝔤 : [𝑥, 𝔤] = {0}} = k e r a d (-)

It is the kernel of the adjoint Lie algebra representation and thus an ideal of $𝔤$ .¹

Properties

Any linear subspace of $𝔷 (𝔤)$ is an ideal of $𝔤$ , called a Central ideal.

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

It is otherwise easy to convince oneself that this is an ideal, since it “absorbs” all elements by sending them to zero. ↩

Centre of a Lie algebra

Properties

Footnotes