nLab
Killing form

Idea

The Killing form or Cartan-Killing form is a binary invariant polynomial that is present on any finite-dimensional Lie algebra.

Definition

Given a finite-dimensional $k$-Lie algebra $𝔤$ its Killing form $B:𝔤\otimes 𝔤\to k$ is the symmetric bilinear form given by the formula

$$B(x,y)=\mathrm{tr}(\mathrm{ad}(x)\mathrm{ad}(y))$$B(x,y) = tr(ad(x)ad(y))

where $\mathrm{ad}(x)=[x,-]:𝔤\to 𝔤$ is the $\mathrm{ad}$-operator giving the adjoint representation $\mathrm{ad}:𝔤\to \mathrm{Der}(𝔤)$.

In terms of a basis: if $\{t_a\}$ is a basis for $𝔤$ and $\{C^a{}_{bc}\}$ the structure constants of the Lie algebra in this basis (defined by $[t_a,t_b]={\sum }_c{C}_{ab}^c{t}_c$), then

$$B(t_a,t_b)=\sum _{c,d}{C}^c{}_{ad}{C}_{bc}^d.$$B(t_a, t_b) = \sum_{c,d} C^c{}_{a d} C^{d}_{b c} \,.

Properties

The Killing form is am invariant polynomial in that

$$B([x,y],z)=B(x,[y,z])$$B([x,y],z)=B(x,[y,z])

for all $x,y,z\in 𝕘$. This follows from the cyclic invariance of the trace],

For complex Lie algebras, nondegeneracy of the Killing form is equivalent to semisimplicity of $𝔤$. For simple complex Lie algebras, any invariant nondegenerate symmetric bilinear form is proportional to the Killing form.

Generalizations

Sometimes one considers more generally a Killing form $B_{\rho }$ for a more general faithful finite-dimensional representation $\rho$, $B_{\rho }(x,y)=\mathrm{tr}(\rho (x)\rho (y))$. If the Killing form is nondegenerate and $x_1,\dots ,x_n$ is a basis in $L$ with $x_1^{*},\dots ,x_n^{*}$ the dual basis of ${𝔤}^{*}$, with respect to the Killing form for $\rho$, then the canonical element $r={\sum }_i{x}_i\otimes x_i^{*}$ defines the Casimir operator? $C(\rho )=(\rho \otimes \rho )(r)$ in the representation $\rho$; regarding that the representation is faithful, if the ground field is $ℂ$, by Schur's lemma $C(\rho )$ is a nonzero scalar operator. Instead of Casimir operators in particular faithful representations it is often useful to consider an analogous construction within the universal enveloping algebra, the Casimir element? in U(\mathfrak{g}).