lecture notes5 - MATH 223A NOTES 2011 LIE ALGEBRAS 17 5...

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MATH 223A NOTES 2011 LIE ALGEBRAS 17 5. Semisimple Lie algebras and the Killing form This section follows Procesi’s book on Lie Groups. We will define semisimple Lie algebras and the Killing form and prove the following. Theorem 5.0.9. The following are equivalent for L a finite dimensional Lie algebra over any subfield F C . (1) L is semisimple. (2) L has no nonzero abelian ideals. (3) The Killing form of L is nondegenerate. (4) L is a direct sum of simple ideals. 5.1. Definition. First we observe that the sum of two solvable ideals I, J in L is solvable. This follows from the fact that I and ( I + J ) /I = J/ ( I J ) 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 solvable ideal. Proposition 5.1.2. L is semisimple i ff 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 [ x [ JJ ]] [[ xJ ] J ] + [ J [ xJ ]] [ JJ ] We have D k J = 0. So D 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 form is clearly symmetric: κ ( x, y ) = κ ( y, x ). The Killing form is also “asso- ciative”: 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 .
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