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lecture notes2 - MATH 223A NOTES 2011 LIE ALGEBRAS 5 2...

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MATH 223A NOTES 2011 LIE ALGEBRAS 5 2. Ideals and homomorphisms 2.1. Ideals. Defnition 2.1.1. An ideal in a Lie algebra L is a vector subspace I so that [ LI ] I .In other words, [ ax ] I for all a L, x I . Example 2.1.2. (1) 0 and L are always ideals in L . (2) If L is abelian then every vector subspace is an ideal. (3) [ LL ] is an ideal in L called the derived algebra of L . Proposition 2.1.3. Every 2-dimensional Lie algebra contains a 1-dimensional ideal. Proof. As we saw last time, there are only two examples of a 2-dimensional Lie algebra: Either the basis elements commute, in which case L is abelian, or they don’t commute, in which case [ LL ] is 1-dimensional. In both cases, L has a 1-dimensional ideal. ° Proposition 2.1.4. The kernel of an homomorphism of Lie algebras ϕ : L L ° is an ideal in L . (The image is a subalgebra of L ° .) Conversely, for any ideal I L , L/I is a Lie algebra and I is the kernel of the quotient map L L/I . Proof. If x ker ϕ and a L then ϕ [ ax ]=[ ϕ ( a ) ϕ ( x )] = [ ϕ ( a )0] = 0. So, [ ax ] ker ϕ . Conversely, if I L is an ideal, a L, x I then [( a + I )( x + I )] [ ax ]+[ aI Ix II ] [ ax ]+ I So, the bracket is well-deFned in L/I ,[ ϕ ( a ) ϕ ( x )] = ϕ [ ax ] and I =ker ϕ . ° Proposition 2.1.5. Any epimorphism (surjective homomorphism) of Lie algebras ϕ : L L ° gives a 1-1 correspondence between ideals I ° in L ° and ideals I of L containing ker ϕ . Proof. The correspondence is given by I ° = ϕ ( I ). This is an ideal since [ L ° ,I ° ]= [ ϕ ( L ) ( I )] = ϕ [ LI ] ϕ ( I )= I ° and I = ϕ 1 ( I ° ) which is an ideal since it is the kernel of L L ° L ° /I ° . ° Example 2.1.6. The center of a Lie algebra L is deFned to be Z ( L { x L | [ xL ]=0 } .
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lecture notes2 - MATH 223A NOTES 2011 LIE ALGEBRAS 5 2...

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