---- > [!definition] Definition. ([[nilpotent element of a semisimple Lie algebra]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] [[Lie algebra]] over $\mathbb{C}$, $x \in \mathfrak{g}$. Since $\mathfrak{g}$ is [[semisimple Lie algebra|semisimple]], $x$ admits a [[Jordan-Chevalley decomposition of an element in a semisimple Lie algebra|Jordan decomposition]] $x=x_{s}+x_{n}.$ We call $x$ **nilpotent** if in fact $x=x_{n}$ with $x_{s}=0$, i.e., its [[adjoint representation]] $\text{ad }x$ is [[nilpotent linear operator|nilpotent]] as a [[linear operator]]. ^definition ---- #### ---- #### 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 > ```