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Unformatted text preview: nce κ = 0 on J ∩ J ⊥ which, by Cartan’s criterion, would imply that JC is
solvable, so J would be solvable.
Since κ is nondegenerate, we have dim J + dim J ⊥ = dim L. Therefore, L = ⊕ J ⊥ .
By induction, J ⊥ is a product of simple ideals. So, L = J1 ⊕ J2 ⊕ · · · ⊕ Jn ∼ Ji . To
prove uniqueness of this decomposition, suppose that I is another minimal ideal. Then
I = [IL] = [IJ1 ] ⊕ [IJ2 ] ⊕ · · · ⊕ [IJn ] One of these summands must be nonzero. Say [IJi ] ⊆ I ∩ Ji = 0. Then I = Ji . Corollary 5.3.2. Let L be a ﬁnite dimensional Lie algebra over F ⊆ C. Then L is
semisimple iﬀ it is a product of simple idea...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11