A continuous group can have generators => repeated application can build all members of the group.

The group generators form an algebra called a lie algebra under commutator.
[Xi,Xj]=Sum(k=1,n,cijk Xk)

Xi,Xj generators of G => [Xi,Xj] also a generator.

X1,..,Xn generators of a compact group => an arbitrary element of the group can be written as exp(-i e*x) where e=vector of stuctures and x are vectors.

Example: SO(2)

Generator:

D(dth)=

|1 -dth|

|dth 1 |

+O(dth^2)

=Id-idthJ+O(dth^2)

where J=

| 0 -i |

| i 0 |

Note that adjoint(J)=J.

D unitary => J hermitian.

dD/dth=-iJD(th)

D(0)=Id.

D(th)=e^-ithJ=Sum(n=0,infinity,(-ith)^n/n! * J^n)=cos(th)I-isin(th)J=SO(2)

Example: SU(2)

looking for 1 parameter subgropu g(t) in SU(2) s.t

- g(t1)g(t2)=g(t1+t2)
- g(t)=e^(-iXt)

t must satisfy:

- t =~ real line ~ T1 (only happens with non-compact groups)
- \/ t =~ a circle ~ U(1)

let g1(t)=

|cos t/2 -isin t/2|

|-isin t/2 cos t/2|

0<=t<=4Pi

=> X1 = id/dt (g1(t)) at t=0.

= .5 * s1 where s1 =

| 0 1 |

| 1 0 |

let g2(t)=

|cos t/2 -sin t/2|

|-sin t/2 cos t/2|

0<=t<=4Pi

=> X2 = id/dt (g2(t)) at t=0.

= .5 * s2 where s2 =

| 0 -i|

| i 0 |

let g3(t)=

|e^(-it/2) 0 |

| 0 e^(it/2)|

0<=t<=4Pi

=> X3 = id/dt (g3(t)) at t=0.

= .5 * s3 where s3 =

| 1 0 |

| 0 -1|

In general, gj(t)= e^(-.5i*sj*t)=3 subgroups of SU(2)

[Xi,Xj]=i*eijk*Xk where eijk are structure constants of SU(2) Lie algebra. [si,sj]=2i*eijk*sk

source

jl@crush.caltech.edu index