[[Lie algebra]]
# Centre of a Lie algebra

The **centre** $\mathfrak{z} (\mathfrak{g})$ of a [[Lie algebra]] $\mathfrak{g}$ is a [[Lie algebra ideal]] consisting of elements which annihilate all elements in the Lie bracket, #m/def/lie i.e.
$$
\begin{align*}
\mathfrak{z} (\mathfrak{g}) = \{ x \in \mathfrak{ g} : [x, \mathfrak{g}] = \{ 0 \} \} = \ker \ad_{(-)}
\end{align*}
$$
It is the [[Kernel of a Lie algebra homomorphism|kernel]] of the [[adjoint Lie algebra representation]]
and thus an [[Lie algebra ideal|ideal]] of $\mathfrak{g}$.[^alt]

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

## Properties

- Any linear subspace of $\mathfrak{z}(\mathfrak{g})$ is an ideal of $\mathfrak{g}$, called a [[Central ideal]].

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