Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

The Killing form was essentially introduced into Lie algebra theory by Élie Cartan (1894) in his thesis. The name "Killing form" first appeared in a paper of Armand Borel in 1951, but he stated in 2001 that he doesn't remember why he chose it. Borel admits that the name seems to be a misnomer, and that it would be more correct to call it the "Cartan form".Wilhelm Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra is invariant under the adjoint group, from which it follows that the Killing form (i.e. the degree 2 coefficient) is invariant, but he did not make much use of this fact. A basic result Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.

Consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad(x) (also written as ad_x) of g with the help of the Lie bracket, as

Now, supposing g is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form

with values in K, the Killing form on g.

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