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
Canonical name	KillingForm
Date of creation	2013-03-22 13:15:47
Last modified on	2013-03-22 13:15:47
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	8
Author	bwebste (988)
Entry type	Definition
Classification	msc 17B05