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Then i il ij1 ij2 ijn one of these summands

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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 ⊥ . 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.

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