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Killing form - Wikipedia

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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.[1] The Killing form was essentially introduced into Lie algebra theory by Élie Cartan (1894) in his thesis. In a historical survey of Lie theory, Borel (2001) 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.[2] Some other authors now employ the term "Cartan-Killing form".[citation needed] 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 w

Killing form - Wikipedia Jump to content From Wikipedia, the free encyclopedia Symmetric bilinear form in mathematics Lie groups and Lie algebras Classical groups General linear GL( n ) Special linear SL( n ) Orthogonal O( n ) Special orthogonal SO( n ) Unitary U( n ) Special unitary SU( n ) Symplectic Sp( n ) Simple Lie groups Classical A n B n C n D n Exceptional G 2 F 4 E 6 E 7 E 8 Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebr

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