What is U(1) Lie Group? | Abelian Lie Group | Unit Circle

[Matterwave]

I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by e^{i\theta}? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

 

[tiny-tim]

U(1) … is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yup!

 

[maverick_starstrider]

I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by e^{i\theta}? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yes, it's not very impressive. Though when we say a U(1) gauge theory we mean the physics is invariant to a LOCAL U(1) gauge transformation (i.e exp(i theta(x)), not just a global one (i.e. exp(i* constant)). However, the reason we label such a simple thing is because we also then establish theories which are invariant under SU(2) and SU(3) local gauge transformations which gives us the weak and strong force physics.

 

[Matterwave]

Ok, last question. Is U(1) identical (homomorphic or diffeomorphic?) to SO(2) then?