---- > [!definition] Definition. ([[invariant bilinear form on a Lie algebra representation]]) > Let $V$ be a [[Lie algebra representation|representation]] of a [[Lie algebra]] $\mathfrak{g}$. We say a [[bilinear map|bilinear map]] $\langle -,- \rangle: V\times V \to \mathfrak{g} $ is **invariant** if $\langle x \cdot v, w \rangle=-\langle v, x \cdot w \rangle$ for all $v,w \in V$, $x \in \mathfrak{g}$. ^definition We know from [[The Cartan-Killing Criterion]] that if $\mathfrak{g}$ is semisimple the [[killing form]] $\mathfrak{g} \times \mathfrak{g} \to \mathbb{C}$ is nondegenerate. In general it is symmetric. The computation $\begin{align} \kappa(x \cdot v, w) & = \kappa([x,v], w) \\ & = -\kappa([v,x], w) \\ & = - \kappa(v, [x,w]) \end{align}$ (using the general properties of [[trace form|trace forms]] and skew-symmetry of the Lie bracket) shows it is invariant. ---- #### ---- #### 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 > ```