---- > [!definition] Definition. ([[inner derivation of a Lie algebra]]) > Let $\mathfrak{g}$ be a [[Lie algebra]]. For all $x \in \mathfrak{g}$, the [[adjoint representation]] gives a map $\text{ad }x:\mathfrak{g} \to \mathfrak{g}$ that is a [[derivation]] of $\mathfrak{g}$, called an **inner derivation**. > The set $\text{Inn }\mathfrak{g}$ all inner derivations is an [[ideal]] of $\text{Der }\mathfrak{g}$. ^definition ---- #### > [!justification] > That $\text{ad }x$ is indeed a [[derivation]] follows from a straightforward application of skew-symmetry and the Jacobi identity We'll use the notation $[y,z]=[yz]$ and $\text{ad }x=\text{ad}_{x}$. > We have $\begin{align} \text{ad}_{x}( [yz] ) &= [x [yz]] \\ & = -[y[zx]] - [z[xy]] \\ &= [y[xz]] + [ [xy]z ] \\ &= [y \ \text{ad}_{x}(z)] + [\text{ad}_{x}(y)z] . \end{align}$ > Next we need to show that $\text{Inn }\mathfrak{g}$ is indeed an [[ideal of a Lie algebra|ideal]] of the [[Lie subalgebra]] $\text{Der }\mathfrak{g}$ of $\mathfrak{g}$. It is of course a [[linear subspace]] of $\mathfrak{g}$ because $\text{ad}_{x}+\text{ad}_{y}=\text{ad}_{x+y} \in \text{Inn }\mathfrak{g}$ and $c \text{ ad}(x)=\text{ad}_{cx}$ for all $c \in \mathbb{F}$, $x,y \in \mathfrak{g}$, by [[bilinear map|bilinearity]] of $[-,-]$. So we just have to show stability under the [[Lie algebra representation|action]]. Let $\delta \in \text{Der }\mathfrak{g}$ and $\text{ad}_{x} \in \text{Inn }\mathfrak{g}$. We claim $[\delta, \text{ad}_{x}]=\text{ad}_{\delta(x)}$. Indeed, $\begin{align} [\delta, \text{ad}_{x}](y) &= \delta \ \text{ad}_{x}(y) - \text{ad}_{x}\delta(y) \\ &= \delta([x,y]) - [x, \delta(y)] \\ &= [x , \delta(y)] + [\delta(x),y] - [x, \delta(y)] \\ &= [\delta(x), y] \\ &= \text{ad}_{\delta(x)}(y) \end{align}$as claimed. ^justification ---- #### 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 > ```