Contents

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 $\mathfrak{g}$ its Killing form $B:\mathfrak{g}\otimes \mathfrak{g}\to k$ is the symmetric bilinear form given by the formula

$B(x,y) \,=\, tr\big( ad(x) \circ ad(y) \big) \,,$

where

$ad(x) \,\coloneqq\, [x,-] \,:\, \mathfrak{g} \longrightarrow \mathfrak{g}$

is the linear map given by the adjoint action of $x$, hence the value on $x$ of the adjoint representation $ad \colon \mathfrak{g} \to Der(\mathfrak{g})$.

If $\{t_a\}$ is a linear basis for $\mathfrak{g}$ and $\{C^a{}_{b c}\}$ are the structure constants of the Lie algebra in this basis (defined by $[t_a, t_b] = \sum_c C^c_{a b} t_c$), then

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

Properties

The Killing form is am invariant polynomial in that

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

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

For complex Lie algebras $\mathfrak{g}$, nondegeneracy of the Killing form (i.e. being the metric making $\mathfrak{g}$ a metric Lie algebra) is equivalent to semisimplicity of $\mathfrak{g}$.

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 $\rho$, $B_\rho(x,y) = tr\big(\rho(x)\rho(y)\big)$. If the Killing form is nondegenerate and $x_1,\ldots,x_n$ is a basis in $L$ with $x_1^*,\ldots,x_n^*$ the dual basis of $\mathfrak{g}^*$, 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 $\mathbb{C}$, 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 $U(\mathfrak{g})$.