---- > [!definition] Definition. ([[centralizer of an element of a Lie algebra]]) > Given a [[Lie algebra]] $\mathfrak{g}$ and $x \in \mathfrak{g}$, the **centralizer** $\mathfrak{z}_{\mathfrak{g}}(x)$ of $x$ in $\mathfrak{g}$ is the [[Lie subalgebra]] $\mathfrak{z}_{\mathfrak{g}}(x):=\{ y \in \mathfrak{g}: [x,y] = 0 \}.$ ^definition > [!basicexample] > Consider the [[special linear Lie subalgebra]] $\mathfrak{sl}_{n}(\mathbb{C})$. Let $e=\begin{bmatrix}0 & 1 \\ 0 &0\end{bmatrix} \in \mathfrak{sl}_{2}(\mathbb{C})$ and let $\mathfrak{g}=\mathfrak{sl}_{3}(\mathbb{C})$. Consider the [[Lie algebra homomorphism]] $\psi:\mathfrak{sl}_{2}(\mathbb{C}) \to \mathfrak{sl}_{3}(\mathbb{C})$ given by $\psi(\begin{bmatrix} > a & b \\ > c & d > \end{bmatrix}) := \begin{bmatrix} > a & b & 0 \\ > c & d & 0 \\ > 0 & 0 & 0 > \end{bmatrix}.$ > Post-composing with the [[adjoint representation]] of $\mathfrak{sl}_{3}(\mathbb{C})$ gives a representation $\text{ad} \circ \psi: \mathfrak{sl}_{2}(\mathbb{C}) \to \mathfrak{gl}\big( \mathfrak{sl}_{3}(\mathbb{C}) \big)$ > of $\mathfrak{sl}_{2}(\mathbb{C})$. > > *Question.* What is $\dim \mathfrak{z}_{\mathfrak{g}}\big( \psi(e) \big)$? > > We have $\psi(e)=\begin{bmatrix} > 0 & 1 & 0 \\ > 0 & 0 & 0 \\ > 0 & 0 & 0 > \end{bmatrix}.$ > ![[CleanShot 2024-10-23 at [email protected]]] > > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```