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 $\mathfrak{g}$ 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 an 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, or compact forms of real Lie algebras, any invariant nondegenerate symmetric bilinear form is proportional to the Killing form. Indeed, one often uses a normalisation of the Killing form on a simple Lie algebra that makes it a positive definite inner product, such that the algebra elements corresponding to long roots (under the isomorphism $\mathfrak{g}^*\simeq \mathfrak{g}$ induced by the original Killing form) have length $\sqrt{2}$. Both the Killing form and this normalised version give a simple Lie algebra a canonical structure of a metric Lie algebra, one positive definite, and one negative definite. This normalised Killing form is sometimes called, following Pressley and Segal, the basic inner product on the Lie algebra.

Here are the basic inner products for the compact forms of the simple real matrix Lie algebras, from (Wang–Ziller 1985, page 583):

• $\mathfrak{su}(n)$: $\langle X,Y\rangle = -tr_{\mathbb{C}}(XY)$.
• $\mathfrak{so}(3)$: $\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/4$.
• $\mathfrak{so}(n)$, $n\geq 5$: $\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/2$.
• $\mathfrak{sp}(n)$: $\langle X,Y\rangle = -Tr(XY) = -2\Re tr_{\mathbb{H}}(XY)$.

The trace $tr_{k}(\cdot)$ here is the ordinary trace for matrices over the division ring $k$, and the reduced trace $Tr(\cdot)$ for $n\times n$ quaternionic matrices is the composite of the embedding into $2n\times 2n$ complex matrices (thinking of $\mathbb{H}\simeq \mathbb{C}\oplus j\mathbb{C}$), and the ordinary trace of complex matrices.

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 $U(\mathfrak{g})$.

References

For a general discussion see:

• Wikipedia, Killing form

Careful discussion around normalisation and related data is in:

• McKenzie Y. Wang and Wolfgang Ziller, On normal homogeneous Einstein manifolds, Annales scientifiques de l’École Normale Supérieure, Série 4 18 (1985), no. 4, 563–633.