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Unformatted text preview: are both solvable.
Deﬁnition 5.1.1. The solvable radical Rad L of L is deﬁned to be the sum of all solvable
ideals. A Lie algebra is semisimple if its solvable radical is zero, i.e., if it has no nonzero
Proposition 5.1.2. L is semisimple iﬀ L has no nonzero abelian ideals.
Proof. If L is semisimple then it has no abelian ideals. Conversely, if L is not semisimple,
then L has a solvable ideal J . Then DJ = [JJ ] is also an ideal in L since
k We have D J = 0. So D [x[JJ ]] ⊆ [[xJ ]J ] + [J [xJ ]] ⊆ [JJ ] k −1 J is a nonzero abelian ideal in L. 5.2. Killing form. The Killing form κ : L × L → F is deﬁned by
κ(x, y ) = Tr(ad x ad y ) The Killing...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11