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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.
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- Fall '11