20a - Lecture 21 Recall that the transporter of a to b in g...

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Lecture 21 Recall that the transporter of a to b in g (denoted by ( b : a )) was de±ned to be the set f z 2 g : ad z ( a ) ± b g . De±nition 1. Let h be a subalgebra of g . Its centralizer is Z g ( h ) = (0 : h ) and its normalizer is N g ( h ) = ( h : h ) Note that both of these are subalgebras of g . In fact, h sits inside of N g ( h ) as an ideal (because [ h : h ] ± h ). Furthermore, N g ( h ) is the largest subalgebra of g containing h as an ideal. Example 1. Consider sl 2 ± sl 3 as sl 2 = ±² A 0 0 0 ³ : A 2 sl 2 ´ If ² W x y T z ³ 2 sl 2 , then ²² A 0 0 0 ³ ; ² W x y T z ³³ = ² [ A;W ] Ax ² y T A 0 ³ Therefore C sl 3 ( sl 2 ) = 8 < : 2 4 ² 1 2 z 0 0 0 ² 1 2 z 0 0 0 z 3 5 : z 2 F 9 = ; ³ F and N sl 3 ( sl 2 ) = ±² W 0 0 T z ³ : W 2 gl 2 ´ ³ gl 2 where z = ² Tr ( W ) . 1
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De±nition 2. The derived series of g is ( g ( n ) : n ± 0) de±ned by g (0) = g and (inductively) g ( n +1) = [ g ( n ) ; g ( n ) ] Notice that each g ( n ) is an ideal in g , and that g = g (0) ² g (1) ² g (2) ² ::: is a decreasing sequence of ideals. De±nition 3. g is solvable if and only if g ( n ) = 0 for some n . Example 2. Any abelian algebra is solvable, with n = 1 . Example 3. 2-dimensionality implies solvability since a nonabelian g has basis a x;y with [ x;y ] = x . This implies that g (1) = F x , which is abelian since it’s 1-dimensional. This in turn implies that g (2) = 0 .
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20a - Lecture 21 Recall that the transporter of a to b in g...

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