---- > [!definition] Definition. ([[adjoint representation]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] over a [[field]] $\mathbb{F}$. The map $\begin{align} \rho: \mathfrak{g}& \to \mathfrak{gl}(\mathfrak{g}) \\ x & \mapsto [x, -] \end{align}$ defines a [[Lie algebra representation]] of $\mathfrak{g}$, called the **adjoint representation**. The map $[x, -]: \mathfrak{g} \to \mathfrak{g}$ is sometimes denoted $\text{ad}_{x}: \mathfrak{g} \to \mathfrak{g}$, and is defined as $\text{ad}_{x}(y):=[x,y]$. ^definition > [!justification] > We need to check that this is indeed a [[Lie algebra representation]]. [[linear map|Linearity]] follows from the fact that Lie brackets are [[bilinear map|bilinear]] and $\text{ad}_{x}$ is just a Lie bracket with one argument fixed. To check that the bracket is preserved, we need to show that for all $x,y \in \mathfrak{g}$, for all $z \in \mathfrak{g}$, that $\overbrace{\text{ad}_{x}(\text{ad}_{y}(z))-\text{ad}_{y}(\text{ad}_{x}(z))}^{[\text{ad}_{x}, \text{ad}_{y}](z)}=\text{ad}_{[x,y]_{\mathfrak{g}}}(z).$ This condition is the same as $[x, [y,z]] - [y, [x,z]]=[[x,y], z].$ Or, rearranging and using that the Lie bracket [[alternating multilinear map|alternates]], the condition reads$[x, [y,z]] + [y,[z,x]] + [ z, [x, y]]=0$ which is true by the Jacobi identity. ^justification > [!basicexample] > See [[Lie algebra representation]]. ^basic-example ---- #### ---- #### 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 > ```