GL == general linear group.

GL(1,C)=g(t,th)=e^(t+ith) 0<=th<2Pi, -infinity<t<infinity

SO(2) is a subgroup given by t=0.

th=0 is another subgroup, e^t=GL(1,R).

GL is non-compact. Integral(-infinity,infinity,dt)=infinity

Irreps of GL are 1-d.

Da(t)=e^at (a is complex)

Some of the irreps are not unitary. a=-il for l in R are the unitary irreps.

GL(n,C) is the most general lie group. All other lie groups are sub groups of it. Dim GL(n,C) = n^2 (complex dimension)

Characters:

X(ab)ij=delta(a,i)delta(b,j) => characters linearly independent.

=> lie algebra with [X(ab),X(gd)]=delta(b,g)Xad-delta(a,d)Xbg

Note that:

Sum(a,Xaa)=Id

GL(n,C) is related to SL(n,C) in the following way:
Let X be a general matrix of GL.

define X~=Xab-1/n*delta(a,b)Id => Tr(X~ab)=0

source

jl@crush.caltech.edu index

lie_group

tensor

SL