---- > [!definition] Definition. ([[killing form]]) > The **killing form** is the [[trace form]] of the [[adjoint representation]]: $\kappa(x,y):=\text{Tr}\big(\text{ad}(x) \ \text{ad}(y)\big).$ ^definition > [!basicproperties] > - [[the killing form includes into the ambient Lie algebra]] > - The killing form is a [[symmetric multilinear map|symmetric form]] (because trace is commutative) > - Recall how [[trace form]] and [[Lie algebra|Lie bracket]] interact > - By Property 3 in [[trace form]], $\kappa(-,-)$ is [[nondegenerate bilinear form|nondegenerate]] whenever $\mathfrak{g}$ is [[semisimple Lie algebra|semisimple]] (in char 0 as usual). [[The Cartan-Killing Criterion|The Cartan-Killing Criterion for semisimplicity]] says that this in fact *characterizes* semisimplicity. ^properties > [!basicexample] Example. (Killing form for $\mathfrak{sl}_{2}(\mathbb{C})$) > Take as [[basis]] of the [[special linear Lie subalgebra]] $\mathfrak{sl}_{2}(\mathbb{C})$ $e,f,h$ [[special linear Lie subalgebra#^basic-example|as defined here]]. [[Lie algebra representation|Recall]] that in this basis, one has $\text{ad}(e)=\begin{bmatrix} 0 & -2 & 0 \\ & 0 & 1 \\ & & 0 \end{bmatrix}, \ \text{ad}(h)=\begin{bmatrix} 2 \\ & 0 \\ & & -2 \end{bmatrix}, \ \text{ad}(f) = \begin{bmatrix} 0 \\ -1 & 0 \\ 0& 2 & 0 \end{bmatrix}.$ [[matrix of a bilinear form|The gram matrix]], obtained by computing $\kappa(e,e),\kappa(e,f),\dots$ is found to be (basis is ordered $e,h,f$) $\begin{bmatrix} 0 & 0 & 4 \\ 0 & 8 & 0 \\ 4 & 0 & 0 \end{bmatrix}.$ > > Note that this matrix is [[inverse matrix|nonsingular]], hence $\kappa$ is a [[nondegenerate bilinear form]]... this turns out to characterize [[semisimple Lie algebra|semisimple Lie algebras]] (recall that $\mathfrak{sl}_{2}(\mathbb{C})$ is semisimple because it is [[simple Lie algebra|simple]], as its [[adjoint representation]] is an [[irreducible Lie algebra representation]].) ^basic-example ---- #### ---- #### 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 > ```