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Concept
Killing form
Summary
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. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.
The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term "Cartan-Killing form". At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are 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 the fact. A basic result that 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 over a field K. Every element x of defines the adjoint endomorphism ad(x) (also written as adx) of with the help of the Lie bracket, as
Now, supposing 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 .
The following properties follow as theorems from the above definition.
The Killing form B is bilinear and symmetric.
The Killing form is an invariant form, as are all other forms obtained from Casimir operators. The derivation of Casimir operators vanishes; for the Killing form, this vanishing can be written as
where [ , ] is the Lie bracket.
If is a simple Lie algebra then any invariant symmetric bilinear form on is a scalar multiple of the Killing form.
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