---- > [!definition] Definition. ([[trace form]]) > Let $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ be a [[Lie algebra representation]] of a [[Lie algebra]] $\mathfrak{g}$. the **trace form of $\rho$** is the [[bilinear map|bilinear form]] $\begin{align} \langle -,- \rangle_{V}: \mathfrak{g \times \mathfrak{g}} \to \mathbb{C} \\ (x,y) \mapsto \text{Tr}\big( \rho(x) \ \rho(y) \big), \end{align}$ where $\text{Tr}$ denotes the [[trace of a linear operator]]. ^definition > [!basicproperties] > - The trace form is [[symmetric multilinear map|symmetric]] > - The trace form and Lie bracket have the following relationship: $([x,y],z)_{V}=(x, [y,z])_{V}$ > - If $\mathfrak{g}$ is [[semisimple Lie algebra|semisimple]] and $\rho$ is [[faithful Lie algebra representation|faithful]], then the trace form is [[nondegenerate bilinear form|nondegenerate]] ^properties - [ ] First two follow from trace(ab)=trace(ba)) Third one: Suppose $\rho$ is [[faithful Lie algebra representation|faithful]], so that it embeds $\mathfrak{g}$ as a [[Lie subalgebra]] of $\mathfrak{gl}(V)$. (Note we can comfortably suppress $\rho$ from the notation.) Let $\mathfrak{k}$ be the kernel of $(-,-)_{V}$, $\mathfrak{k}=\{ y \in \mathfrak{g} : (x,y)_{V}=0 \text{ for all } x \in \mathfrak{g}\}.$ If $x \in \mathfrak{k}$ and $y \in [\mathfrak{k}, \mathfrak{k}] \subset \mathfrak{k}$, then $\text{Tr}(xy)=(x, y)_{V}=0$. By [[Cartan's Trace Theorem]], $\mathfrak{k}$ must be [[derived and central series of a Lie algebra|solvable]]. By [[The Cartan-Killing Criterion|The Cartan-Killing Criterion for semisimplicity]], then, $\mathfrak{k}=0$. ---- #### ---- #### 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 > ```