This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 03 Example 1. A 2dimensional, 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 3dimensional, 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 3dimensional, 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 1dimensional 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...
View
Full
Document
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.
 Summer '10
 Staff
 Algebra

Click to edit the document details