---- > [!definition] Definition. ([[Casimir element]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] [[Lie algebra]] over $\mathbb{C}$, $V$ a [[vector space]], and $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ a [[faithful Lie algebra representation|faithful]] [[Lie algebra representation|representation]] of $\mathfrak{g}$. Since $\rho$ is [[faithful Lie algebra representation|faithful]] and $\mathfrak{g}$ is semisimple, recall the [[trace form]] $(-,-)_{V}$ is [[nondegenerate bilinear form|nondegenerate]]. > Let $x_{1},\dots,x_{m}$ be a [[basis]] of $\mathfrak{g}$ and let $y^{1},\dots,y^{m}$ be the [[dual basis]] with respect to [[trace form|trace form of]] $\rho$: $(x_{i}, y^{j})_{V}=\delta^{i}_{j}$ for all $i,j$. Then the [[morphism]] $\Omega_{\rho}=\sum_{i=1}^{m} \rho(x_{i}) \rho(y_{i}) \in \text{End }V$ of the [[Lie algebra representation|representation]] $V$ is called the **Casimir element** associated to $\rho$. > It turns out that $\Omega_{\rho}$ does not depend on the choice of [[basis]] of $\mathfrak{g}$. We won't need or prove this, but it justifies leaving this choice implicit in the notation. ^definition > [!justification] > - [ ] Need to show $\mathfrak{g}$-equivariance... ^justification > [!specialization] > A provisional definition of the Casimir element in the context of $\mathfrak{sl}_{2}(\mathbb{C})$ featured in the proof of [[completely reducible|complete reducibility]] for $\mathfrak{sl}_{2}(\mathbb{C})$ . ^specialization [^1]: By [[Cartan's Trace Theorem]], the [[kernel of a bilinear form|kernel]] $\mathfrak{k}$ of $(-,-)_{V}$ is a [[derived and central series of a Lie algebra|solvable]] [[ideal of a Lie algebra|ideal]]: $\mathfrak{k}$ ---- #### ---- #### 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 > ```