p600_02h

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					Lie Generators
Lie Group Operation
   Lie groups are continuous.      The multiplication law is by
     • Continuous coordinate         analytic functions.
       system                         • Two elements x, y
     • Finite dimension               • Consider z = xy
     • Origin is identity
                                    There are N analytic
      x  ( ,  ,  )
             1       N
                                     functions that define the
      y  ( 1 ,  , N )            coordinates.
                                      • Based on 2N coordinates

    z  ( 1 , ,  N )  xy

         f  ( , )
GL as Lie Group
   The general linear groups       Example
    GL(n, R) are Lie groups.         Let x, y  GL(n, R).
     • Represent transformations          • Coordinates are matrix
     • Dimension is n2                      elements minus db
                                               b  xb  d b

                                        Find the coordinates of z=xy.
                                         b  zb  d b  x y b  d b
   All Lie groups are isomorphic
    to subgroups of GL(n, R).            b    d   b  d b   d b
                                         b   b  b  b

                                          • Analytic in coordinates
Transformed Curves
                                   All Lie groups have
                                    coordinate systems.
                                  • May define differentiable
      
     ( 0)         de
               e                      curves
                       (e )
                                   The set x(e) may also form a
  x(0)  e   x(e )  G
                                    group.
                                     • Subgroup g(e)
Single-axis Rotation
    Parameterizations of                          Example
     subgroups may take different                   Consider rotations about the
     forms.                                           Euclidean x-axis.
                                                        • May use either angle or sine
    e 
                g (e1 ) g (e 2 )  g (e1  e 2 )
                                                      The choice gives different
    e  sin                                           rules for multiplication.

                    
        g (e1 ) g (e 2 ) 
                                    
        g (e1[1  e 2 ]1 2  e 2 [1  e1 ]1 2 )
                       2                  2
One Parameter
   A one-parameter subgroup          g (e1 ) g (e 2 )  g (e1  e 2 )
                                                           S1
    can always be written in a
    standard form.                    g (e1 ) g (e 2 )  g[  (e1 , e 2 )]
                                                               
     • Start with arbitrary                        (e ,0)   (0, e )  e 
       represenatation
     • Differentiable function      e   e (e )        e (0)  0
     • Assume that there is a
       parameter                     e (e1  e 2 )   (e (e1 ), e (e 2 ))
                                     de  de      
   The differential equation will       
                                     de               
                                           de e 0 e 2        
                                                             e 2 0
    have a solution.
     • Invert to get parameter       de 
                                           kf (e )
                                     de
Transformation Generator
    The standard form can be used         dg 1          g (e  e )  g (e ) 
                                         g        g lim                         
     to find a parameter a                  de                     e            
     independent of e.
                                                g (e ) g (e  e )  g (e ) g (e ) 
                                          lim                                       
gg 1  e                                                       e                   
                    dg 1     dg 1             g (e )  g (0) 
                       g g         0    lim 
                    de         de                       e          g ( 0)   a
                                                                  
                                                                          
                    dg      dg 1
                        g       g          Solve the differential equation.
                    de       de

                                             dg
    Using standard form                          ag                 g (e )  eea
                                             de
    g 1 (e )  g (e )
                                             The matrix a is an infinitessimal
next                                          generator of g(e)

				
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posted:11/10/2011
language:English
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