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