Why Must the Parameter p be an Integer in U(1) Representations?

[Pietjuh]

Hello, perhaps this is the most dumb question ever, but I don't see why it holds.
I'm looking at the irreducible representations of the Lie group U(1). To find them I considered the irreps of the lie algebra u(1). These irreps are obviously 1 dimensional and are given by f(a i ) = p a i for some real number p. If I now exponentiate this result I find that the irreducible representations of U(1) are given by g( exp(i theta) ) = exp(i p theta). But I read that p must be an integer. I cannot see however why this must be true! :(

Thanks in advance

 

[matt grime]

What is "a i"?

Of course, have you worked out the lie algebra of U(1) correctly? Are you allowing for the fact that you really only want unitary reps, and that even if as claimed the lie algebra is just the complex numbers that this is not a semi-simple lie algebra over C?

 

[George Jones]

For every integer n, exp( 2 n pi i theta) is the same element of U(1) as exp(i theta). Hence, g( exp(i theta)) = g( exp( 2 n pi i theta)) for every integer n.

 

[Pietjuh]

What is "a i"?

Of course, have you worked out the lie algebra of U(1) correctly? Are you allowing for the fact that you really only want unitary reps, and that even if as claimed the lie algebra is just the complex numbers that this is not a semi-simple lie algebra over C?

Let me clarfiy my derivation of the Lie algebra of U(1). Let exp(i theta) be an arbitrary element of U(1). Then the Lie algebra is the tangent space at the identity element, so u(1) is spanned by the basis vector i. This means that u(1) = { ai | a in R }.

 

[matt grime]

Since U(1) is not simply connected the correspondence between representations of U(1) and u(1) fails to hold generally.

 

[George Jones]

Since U(1) is not simply connected the correspondence between representations of U(1) and u(1) fails to hold generally.

I'm not sure what you mean. Because U(1) is not simply connected, any Lie Group that is a covering of U(1) shares a Lie algebra with U(1).

However, U(1) is connected and compact, and therefore the exponential map from u(1) to U(1) is onto.

Isn't this enough foe what Pietjuh wants?

 

[matt grime]

I am not entirely shure what it is pietjuh actually wants, to be honest.

 

[George Jones]

Sorry, Pietjuh, I didn't see my typo until now

For every integer n, exp( 2 n pi i theta) is the same element of U(1) as exp(i theta). Hence, g( exp(i theta)) = g( exp( 2 n pi i theta)) for every integer n.

Cleary, this should be g( exp(i theta)) = g( exp( 2 n pi i + i theta)).

Now do you see why p must be an integer?