# 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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 views: 4 posted: 11/10/2011 language: English pages: 7