22a - Lecture 22 Theorem 1. (Engel) If each element of g is...

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Theorem 1. (Engel) If each element of g is ad-nilpotent, then g is nilpotent. Proof of a particular case : Suppose each x g satisfies ad 2 x = 0. That is, for all x,y g , [ x, [ x,y ]] = 0. Claim g 2 = 0, provided char F 6 = 3. That is, [ x, [ y,z ]] = 0 for all x,y,z g . Given w,z g = [ z, [ z,w ]] = 0. Replace z by x + y . 0 = [ x + y, [ x + y,w ]] = [ x, [ x,w ]] + [ x, [ y,w ]] + [ y, [ x,w ]] + [ y, [ y,w ]] = 0 + [ x, [ y,w ]] + [ y, [ x,w ]] + 0 [ x, [ y,w ]] is skew symmetric in x,y Now by Jacobi, 0 = [ w, [ x,y ]] + [ x, [ y,w ]] + [ y, [ w,x ]] = [ w, [ x,y ]] - [ x, [ w,y ]] - [ w, [ y,x ]] = [ w, [ x,y ]] + [ w, [ x,y ]] + [ w, [ x,y ]] = 3[ w, [ x,y ]] g 2 = 0 provided char F 6 = 3. Remark : In this case it turns out that g 3 = 0. To prove Engel, we use Theorem 2. Let V be a finite dimensional vector space and g gl ( V ) a subalgebra. If each element of g is nilpotent as a linear map V V , then T z g Ker ( z ) 6 = 0 . 1
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

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22a - Lecture 22 Theorem 1. (Engel) If each element of g is...

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