Lie algebra

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, lie i.e.

𝔷(𝔤)={𝑥∈𝔤:[𝑥,𝔤]={0}}=ker⁡ad(−)

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.

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Footnotes

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