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The killing form is also associative xy z x yz

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Unformatted text preview: form is clearly symmetric: κ(x, y ) = κ(y, x). The Killing form is also “associative”: κ([xy ], z ) = κ(x, [yz ]) Proof. Since ad[xy ] = [ad x, ad y ], we have: κ([xy ], z ) = Tr(ad [xy ] ad z ) = Tr([ad x, ad y ]ad z ) = Tr(ad x[ad y, ad z ]) = κ(x, [yz ]) Proposition 5.2.1. The Killing form is invariant under any automorphism ρ of L. Proof. The equation ρ[xy ] = [ρ(x)ρ(y )] for z = ρ(y ) is ρ[x, ρ−1 (z )] = [ρ(x)z ] which can be rewritten as: ad ρ(x) = ρ ◦ ad x ◦ ρ−1 . So, κ(ρ(x), ρ(y )) = Tr(ad ρ(x) ad ρ(y )) = Tr(ρ ◦ ad x ad y ◦ ρ−1 ) = Tr(ad x ad y ) = κ(x, y ). ￿ Lemma 5.2.2. The kernel (also called the radi...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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