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Unformatted text preview: Lecture 04 Definition 1. A derivation of the algebra A is a linear map : A A such that x,y A ( xy ) = ( x ) y + x ( y ) . Theorem 1. The space Der A comprising all derivations of A is a Lie sub- algebra of gl ( A ) . Proof. Der A is a subspace of gl ( A ). It remains (as an easy exercise) to show that if , Der A then [ , ] := - Der A. In particular, Der g is a subalgebra of gl ( g ). Theorem 2. z g ad z Der( g ) Proof. Jacobi (plus alternating): ad z [ x,y ] = [ z, [ x,y ]] =- [ y, [ z,x ]]- [ x, [ y,z ]] = [[ z,x ] ,y ] + [ x, [ z,y ]] = [ad z x,y ] + [ x, ad z y ] . 1 So the image of ad : g gl ( g ) is contained in Der g . Theorem 3. If Der g and z g then [ , ad z ] = ad z . Proof. If w g then [ , ad z ]( w ) = (ad z w )- ad z ( w ) = [ z,w ]- [ z,w ] = [ z,w ] = ad z ( w ) . So ad( g ) is an ideal in Der( g ) (but not necessarily in gl ( g ))....
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- Summer '10