---- Cartan's (Trace) Theorem gives a computational characterization of solvability. > [!theorem] Theorem. ([[Cartan's Trace Theorem]]) > Let $\mathbb{F}$ be a [[field]] that is [[algebraically closed]] and has [[characteristic of a field|characteristic zero]] (e.g., $\mathbb{F}=\mathbb{C}$). Let $V$ be an $\mathbb{F}$-[[vector space]]. Let $\mathfrak{gl}(V)$ denote the [[general linear Lie algebra]] over $V$, and let $\mathfrak{g} \subset \mathfrak{gl}(V)$ be a [[Lie subalgebra]]. > Then $\mathfrak{g}$ is [[derived and central series of a Lie algebra|solvable]] if and only if $\text{Tr}(xy)=0$ for all $x \in \mathfrak{g}$ and $y \in \mathfrak{[g,g]=\mathfrak{g}^{}}^{(1)}$, the [[derived and central series of a Lie algebra|derived subalgebra]]. ^theorem > [!proof]- Proof. ([[Cartan's Trace Theorem]]) > Omitted in our course. ---- #### (Here, $\text{Tr}$ denotes the [[trace of a matrix|trace operation]].) ----- #### 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 > ```