---- > [!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. > - If $\mathfrak{g}$ is the [[Lie algebra]] of a [[compact]] [[Lie group]] $G$, then $\kappa$ is negative-definite modulo the [[center of a Lie algebra|center]]: $\kappa(x, x) \leq 0 \text{ with equality } \iff \operatorname{ad }x=0 (\iff x \in Z(\mathfrak{g})).$ (This condition can help rule out whether a given f.d. real [[Lie algebra]], which we know always [[the category of simply connected Lie groups is equivalent to that of finite-dimensional real lie algebras|integrates to]] a [[simply connected]] [[Lie group]], in fact integrates to a [[compact]] one.) > > > [!proof]- Proof. > > Suppose $\mathfrak{g}=T_{e}G$ is the [[Lie algebra]] of a [[compact]] [[Lie group]] $G$. Let $X \in \mathfrak{g}$. Since $G$ is [[compact]], the [[adjoint representation]] $\operatorname{Ad}:G \to \operatorname{Aut}\mathfrak{g}$ [[compact group representations are unitary|is unitary]], meaning that $\mathfrak{g}$ admits an $\operatorname{Ad}$-[[unitary group representation|invariant inner product]] so that $\operatorname{Ad}$ corestricts to a [[Lie group|Lie group homomorphism]] $G \to \operatorname{O}(n)$. Thus its [[differential of a smooth map between smooth manifolds|differential]] $\operatorname{ad}: \mathfrak{g} \to \mathfrak{o}(n)$ where $\mathfrak{o}(n)$ is the space of $n \times n$ [[skew-symmetric matrix|skew-symmetric matrices]]. The $\kappa(X,X)=\operatorname{tr}(A ^{2})=\operatorname{tr}(-A^{\top} A)=-\operatorname{tr}(A ^{\top} A)$ for some skew-symmetric matrix $A$. Since $A^{\top }A$ is [[positive semidefinite matrix|positive semidefinite]], its [[eigenvalue|eigenvalues]] are all nonnegative, thus $\kappa(X,X) \leq 0$ in general. > > > > [!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 > ```