---- Let $V$ be a $\mathbb{C}$-[[vector space]] and $\mathfrak{gl}(V)$ denote the [[general linear Lie algebra]] over $V$. > [!theorem] Theorem. ([[Jordan-Chevalley decomposition of an element in a semisimple Lie algebra]]) > **1.** Let $\mathfrak{h}$ be a [[semisimple Lie algebra|semisimple]] [[Lie subalgebra]] of $\mathfrak{gl}(V)$, $x \in \mathfrak{h}$; (uniquely) [[Jordan-Chevalley decomposition of a linear operator|write]] $x=x_{s}+x_{n}$ for $x_{s}$ a [[diagonalizable|semisimple]] [[linear operator]] and $x_{n}$ a [[nilpotent linear operator]]. Then $x_{s},x_{n} \in \mathfrak{h}$. > **2.** Now let $\mathfrak{g}$ be *any* [[semisimple Lie algebra|semisimple]] [[Lie algebra]]. Recall this means the [[adjoint representation]] $\text{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$ is [[faithful Lie algebra representation|an embedding]]. It follows from **1**, then, that any $x \in \mathfrak{g}$ uniquely decomposes as $x=x_{s}+x_{n}$, where $\text{ad }x_{s}$ is [[diagonalizable|semisimple]] and $\text{ad }x_{n}$ is [[nilpotent linear operator|nilpotent]]: $\text{ad}(x)=\text{ad}(x)_{s}+\text{ad}(x)_{n}=\text{ad}(x_{s})+\text{ad}(x_{n})$ This is called the (abstract) **Jordan decomposition of $x$**. > **3.** If $\mathfrak{g}$ is in fact a [[semisimple Lie algebra|semisimple]] [[Lie subalgebra]] of $\mathfrak{gl}(V)$, then there are two 'Jordan decompositions' of $x \in \mathfrak{g}$ to speak of: [[Jordan-Chevalley decomposition of a linear operator|that of]] $x$ as a [[linear operator]], and that (**2**) of $x$ as an element of a semisimple Lie algebra. There is in fact no ambiguity: the two notions agree on this overlap. > >**4.** For any representation $\rho$ of $\mathfrak{g}$ (viewed as any Lie algebra once again) and $x=x_{s}+x_{n} \in \mathfrak{g}$, we have that $\rho(x)=\rho(x_{s})+ \rho(x_{n})$ is the [[Jordan-Chevalley decomposition of a linear operator|Jordan decomposition]] of $\rho(x)$ as a [[linear operator]]. So again, there is no ambiguity. ^theorem > [!proof]- Proof. ([[Jordan-Chevalley decomposition of an element in a semisimple Lie algebra]]) > Not in our course. ---- #### ----- #### 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 > ```