Transcript Lie Generators

Lie Generators
Lie Group Operation

Lie groups are continuous.

• Continuous coordinate
system
• Finite dimension
• Origin is identity
x  ( ,,  )
1
N
y  ( 1 ,, N )
z  ( 1 ,,  N )  xy
   f  ( , )
The multiplication law is by
analytic functions.
• Two elements x, y
• Consider z = xy

There are N analytic
functions that define the
coordinates.
• Based on 2N coordinates
GL as Lie Group

The general linear groups
GL(n, R) are Lie groups.
Example
 Let x, y  GL(n, R).
• Represent transformations
• Dimension is n2
• Coordinates are matrix
elements minus db
b  xb  d b


All Lie groups are isomorphic
to subgroups of GL(n, R).
Find the coordinates of z=xy.
 b  zb  d b  x yb  d b
 b    d   b  d b   d b
 b   b  b  b
• Analytic in coordinates
Transformed Curves


 (0)
 
de
e
• May define differentiable
curves
  (e )
x ( 0)  e
x(e )  G
All Lie groups have
coordinate systems.

The set x(e) may also form a
group.
• Subgroup g(e)
Single-axis Rotation

Parameterizations of
subgroups may take different
forms.
e 
Example
 Consider rotations about the
Euclidean x-axis.
• May use either angle or sine
g (e1 ) g (e 2 )  g (e1  e 2 )
e  sin 
g (e1) g (e 2 ) 
g (e1[1  e 2 ]1 2  e 2 [1  e1 ]1 2 )
2
2

The choice gives different
rules for multiplication.
One Parameter

A one-parameter subgroup
can always be written in a
standard form.
• Start with arbitrary
represenatation
• Differentiable function 
• Assume that there is a
parameter

The differential equation will
have a solution.
• Invert to get parameter
g (e1 ) g (e 2 )  g (Se1 1  e 2 )
g (e1) g (e 2 )  g[ (e1, e 2 )]
 (e ,0)   (0, e )  e 
e   e (e )
e (0)  0
e (e1  e 2 )   (e (e1 ), e (e 2 ))
de  de 


de
de e 0 e 2
de 
 kf (e )
de
e 2  0
Transformation Generator

The standard form can be used
to find a parameter a
independent of e.
gg 1  e
dg 1
dg 1
g g
0
de
de
dg
dg 1
 g
g
de
de
dg 1
 g (e  e )  g (e ) 
g
 g lim 

de
e

 g (e ) g (e  e )  g (e ) g (e ) 
 lim 

e

 g (e )  g (0) 
 lim 
  g (0)  a

e



dg
 ag
de
Using standard form
g 1 (e )  g (e )

next
Solve the differential equation.
g (e )  eea
The matrix a is an infinitessimal
generator of g(e)