Forma di Killing

In matematica, la forma di Killing è una forma bilineare simmetrica che svolge un ruolo fondamentale nella teoria delle algebre di Lie e in quella dei gruppi di Lie. Prende il nome da Wilhelm Killing, che per primo ne diede menzione. Il primo a introdurla nella teoria delle algebre di Lie è stato Élie Cartan.

Definizione

Sia ${\mathfrak {g}}$ un'algebra di Lie di dimensione finita su un campo $K$ . Ogni elemento $x$ di ${\mathfrak {g}}$ definisce l'endomorfismo aggiunto ${\textrm {ad}}(x)$ , definito come

{\textrm {ad}}(x)(y):=[x,y].

Poiché ${\mathfrak {g}}$ è di dimensione finita, la traccia della composizione di due endomorfismi è una forma bilineare simmetrica. La forma di killing $B(x,y)$ su ${\mathfrak {g}}$ è quindi definita come

B(x,y):={\textrm {Tr}}({\textrm {ad}}(x){\textrm {ad}}(y)).

Bibliografia

Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4

Voci correlate

Criterio di Cartan

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