03 - Lecture 03 Example 1. A 2-dimensional, non abelian Lie...

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Unformatted text preview: Lecture 03 Example 1. A 2-dimensional, non abelian Lie algebra. Consider: g = { b a : a,b F } spanned by x = 0 0 1 0 , y = 1 0 0 0 Notice then that [ x,y ] = x Z ( g ) , g gl 2 ( F ) , dim g = 2 , Z ( g ) 6 = { } . Example 2. A 3-dimensional, non abelian Lie algebra, with [ g , g ] Z ( g ) and dim [ g , g ] = 1 . Consider: g = { a b 0 0 c 0 0 0 : a,b,c F } spanned by x = 0 1 0 0 0 0 0 0 0 , y = 0 0 0 0 0 1 0 0 0 , z = 0 0 1 0 0 0 0 0 0 This is a subalgebra of gl 3 ( F ) with the properties [ x,y ] = z Z ( g ) , [ z,x ] = 0 , [ z,y ] = 0 , called the Heisenberg Algebra. 1 Example 3. A 3-dimensional, non abelian Lie algebra, with [ g , g ] * Z ( g ) and dim [ g , g ] = 1 . Consider: g = { b a 0 0 c a,b,c F } spanned by x = 0 0 0 1 0 0 0 0 0 , y = 1 0 0 0 0 0 0 0 0 , z = 0 0 0 0 0 0 0 0 1 This is a subalgebra of gl 3 ( F ) satisfying the properties [ x,y ] = x, [ x,z ] = , [ y,z ] = 0 . It is also isomorphic to the direct sum of the algebra presented at the first example and the 1-dimensional algebra F (look at the splitting of the matrix!). Back to the classification. Dim g = 3 dim [ g , g ] = 2 Claim: [ g , g ] is abelian. Proof. Let { x,y } be a basis for [ g , g ] and pick z g- [ g , g ]. Notice that [ x,y ] [ g , g ] thus [ x,y ] = ax + by for some a,b F Then define X := [ x, ] : g g This is a linear map denoted ad x . Notice that Im ( X ) [ g , g ]. Thus X ( x ) = , X ( y ) = ax + by, X ( z ) = cx + dy . Clearly Trace ( X ) = tr ( X ) = 0+ b +0 = b . Consider the corresponding map Y := [ y, ] : g g Likewise tr ( Y ) =- a . On the other hand notice that if u,v g then [[ u,v ] , ] = [[ u, ] , [ v, ]] = [ u, ] [ v, ]- [ v, ] [ u, ] where is composition of linear maps. Thus: tr [[ u,v ] , ] = tr ([ u, ] [ v, ]- [ v, ] [ u, ]) = 0 2 Then remember that x belongs to [ g , g ]. So x...
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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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03 - Lecture 03 Example 1. A 2-dimensional, non abelian Lie...

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