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This is 3 dimensional with basis x12 x23 x13 and the

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Unformatted text preview: gl(I ) and the kernel of this is the centralizer ZL (I ) of I in L. This is an ideal in L since it is a kernel. Example 2.1.7. Consider the Lie algebra n(3, F ). This is 3-dimensional with basis x12 , x23 , x13 and the only nontrivial bracket is [x12 x23 ] = x13 . Then L = n(3, F ) has the property that [LL] = Z (L) is one-dimensional. Conversely, it is easy to see that any 3-dimensional Lie algebra with [LL] = Z (L) must be isomorphic to n(3, F ). Example 2.1.8. Tr : gl(n, F ) → F is a homomorphism of Lie algebras (with F the abelian Lie algebra) since Tr[xy ] = Tr(xy ) − Tr(yx) = 0 = [Tr(x), Tr(y )]. Therefore, sl(n, F ) is an ideal in gl(n, F ). 6 MATH 223A NOTES 2011 LIE ALGEBRAS Exercise 2.1.9. What is the center of sl(2, F )? A Lie algebra is called simple if it is nonabelian and has no nontrivial proper ideals. Theorem 2.1.10. sl(2, F ) is simple if char F ￿= 2. sl(2, F ) is a key example of a Lie algebra which you need to understand very thoroughly. Proof. Since L = sl(2, F ) is 3-dimensional we just need to show that it has no ideals of dimension 1 and no ideals of dimension 2. T...
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