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Unformatted text preview: emisimple iﬀ its Killing form is nondegenerate.
Proof. Suppose that the Killing form of L is nondegenerate. Then L must be semisimple
since any abelian ideal is contained in the kernel of κ which is zero. Conversely, suppose
that the Killing form of L has a nonzero kernel S .
Let LC = L ⊗F C be the complexiﬁcation of L. Since L ⊆ LC , it is clear that L is abelian
iﬀ LC is abelian. This implies the L is solvable iﬀ LC is solvable since D(LC ) = (DL)C .
One can also see that the Killing form κC of LC is the complexiﬁcation of the Killing form
κ of L. So, the kernel of κC is SC . We know that SC is solvable from the proof of Cartan’s
Theorem. Therefore S is solvable and L is not semisimple.
Example 5.2.6. Let L = sl(2, R). This has basis
10 h= with respect to this basis we have 0 0 −2
ad x = 0 0 0 , ad y = 0 0 2 ,
−1 0 0 Then
0 −1 200
ad h = 0 −2...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11