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# If l is not semisimple then it has a nonzero abelian

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Unformatted text preview: nite dimensional Lie algebra over C. Then L is semisimple iﬀ its Killing form is nondegenerate (its kernel S = 0). Proof. We will show that S ￿= 0 iﬀ L is not semisimple. If L is not semisimple then it has a nonzero abelian ideal. Any such ideal lies in the kernel S . So, S ￿= 0. Conversely, suppose that S ￿= 0. Then Cartan’s criterion shows that the image adL S of S under the adjoint representation adL : L → gl(L) is solvable since κ(x, y ) = Tr(ad x ad y ) = 0 for all x, y ∈ S . Since adL S = S/Z (L), this implies that S is solvable. Therefore, L is not semisimple. ￿ Corollary 5.2.5. Suppose that L is a ﬁnite dimensional Lie algebra over a subﬁeld F of C. Then L is s...
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## This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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