# Killing form

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over a field $k$, and $X,Y\in\mathfrak{g}$. Let $\mathrm{ad}_{X}:\mathfrak{g}\to\mathfrak{g}$ be the adjoint action, $\mathrm{ad}_{X}Y=[X,Y]$.

Then the Killing form on $\mathfrak{g}$ is a bilinear map

 $B_{\mathfrak{g}}:\mathfrak{g}\times\mathfrak{g}\to k$

given by

 $B_{\mathfrak{g}}(X,Y)=\mathrm{tr}(\mathrm{ad}_{X}\circ\mathrm{ad}_{Y}).$

The Killing form is invariant (http://planetmath.org/InvariantFormLieAlgebras) under the adjoint action and symmetric (since trace is symmetric).

Title Killing form KillingForm 2013-03-22 13:15:47 2013-03-22 13:15:47 bwebste (988) bwebste (988) 8 bwebste (988) Definition msc 17B05