lie groups are continuous groups.

Classical Lie group:

- group multiplication = matrix multiplication
- members: S0(n), O(n), GL(n, Complex), GL(n, Real), SL(n, Complex), SL(n, Real), U(n), SU(n)

S0(2) = rotational symmetry of a circle. U(1) = phase matrix. Isomorphic to SO(2). O(2) = reflections and rotations of a circle.

GL(1,C)=g(t,th)=e^(t+ith) 0<=th<2Pi, -infinity<t<infinity
GL(1,R)+ = T = 1-d translation group.

SO(1,1) Isomorphic to T.

SO(n) has n(n-1)/2 parameters, compact

SU(n) has n*n-1 parameters, compact

The group manifold of SU(2) is S3.

source

jl@crush.caltech.edu index

continuous_group

covering_group

GL