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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
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.
- Fall '11