Killing form - Imported Knowlpedia Content

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ (typically $\mathbb R$ or $\mathbb C$ ). Let $\mathrm{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g)$ be the adjoint representation.

Definition (Killing form). The Killing form on $\mathfrak g$ is the symmetric bilinear form

B:\mathfrak g\times \mathfrak g\to \Bbbk,\qquad B(x,y)=\mathrm{tr}(\mathrm{ad}_x\circ \mathrm{ad}_y).

It is ad-invariant and depends functorially on $\mathfrak g$ .

Remarks

Example: $\mathfrak{sl}_2(\mathbb C)$ . With basis

H=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad E=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad F=\begin{pmatrix}0&0\\1&0\end{pmatrix},

one computes (using $\mathrm{ad}_X(Y)=[X,Y]$ ) that

B(H,H)=8,\qquad B(E,F)=4,\qquad B(H,E)=B(H,F)=B(E,E)=B(F,F)=0.

This exhibits nondegeneracy for a simple algebra.

Example: $\mathfrak{sl}_n(\mathbb C)$ . For $X,Y\in \mathfrak{sl}_n(\mathbb C)$ (see sl_n), the Killing form is a scalar multiple of the trace pairing; in the standard normalization,

B(X,Y)=2n\,\mathrm{tr}(XY).

Context. Nondegeneracy of $B$ characterizes semisimplicity (see the nondegeneracy theorem) and underlies criteria such as Cartan's criterion.