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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.
==History and name==
The Killing form was essentially introduced into Lie algebra theory by 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"''.〔Borel, p.5〕 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.〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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