04 - Lecture 04 Definition 1. A derivation of the algebra A...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ))....
View Full Document

Page1 / 4

04 - Lecture 04 Definition 1. A derivation of the algebra A...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online