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Representations of U(1)

  1. Jul 12, 2006 #1
    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
     
  2. jcsd
  3. Jul 12, 2006 #2

    matt grime

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    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?
     
    Last edited: Jul 12, 2006
  4. Jul 12, 2006 #3

    George Jones

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    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.
     
  5. Jul 12, 2006 #4
    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 }.
     
  6. Jul 12, 2006 #5

    matt grime

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    Since U(1) is not simply connected the correspondence between representations of U(1) and u(1) fails to hold generally.
     
  7. Jul 12, 2006 #6

    George Jones

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    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?
     
  8. Jul 12, 2006 #7

    matt grime

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    I am not entirely shure what it is pietjuh actually wants, to be honest.
     
  9. Jul 13, 2006 #8

    George Jones

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

    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?
     
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