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Non-compact form of an algebra.

  1. Jul 2, 2013 #1
    I read that in string theory the Virasoro algebra contains an ##SL(2,R)## subalgebra that is generated by ##L_{-1}, L_{0}, L_{1}##. I read that this is the non-compact form of the ##SU(2)## algebra. Also, that as ##SU(2)## and ##SO(3)## have the same Lie algebra, so do ##SL(2,R)## and ##SO(2,1)##.

    Can someone explain all the above statementes? I understand what a compact group is and I have seen that ##SU(2)## and ##SO(3)## have the same Lie algebra. But, what do the other statements mean?

    Can you also give a more intuitive explanation?

    Thank you very much.
     
  2. jcsd
  3. Jul 2, 2013 #2

    Ben Niehoff

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    What's intuitive to one person might not be to another.

    To obtain a "non-compact form" of a Lie algebra, you multiply some of the generators by ##i##. So if ##X_1, X_2, X_3## have the ##SU(2)## algebra

    [tex][X_1, X_2] = X_3, \qquad [X_2, X_3] = X_1, \qquad [X_3, X_1] = X_2,[/tex]
    then the new set of generators given by

    [tex]\tilde X_1 \equiv i X_1, \qquad \tilde X_2 \equiv i X_2, \qquad \tilde X_3 \equiv X_3[/tex]
    will have the algebra

    [tex][\tilde X_1, \tilde X_2] = -\tilde X_3, \qquad [\tilde X_2, \tilde X_3] = \tilde X_1, \qquad [\tilde X_3, \tilde X_1] = \tilde X_2,[/tex]

    which is the algebra of ##SL(2,\mathbb{R})##.

    A Lie group is "compact" or "non-compact" depending on the eigenvalues of its Cartan-Killing form. If all the eigenvalues are negative, then the group is compact. If some of the eigenvalues are positive, then the group is non-compact. The Killing form of ##SL(2, \mathbb{R})## has signature ##(+, +, -)##, and it has the topology ##R^2 \times S^1##, so you see there are two noncompact directions, and one compact, matching the signs in the Killing form.

    Both ##SU(2)## and ##SL(2,\mathbb{R})## algebras are subalgebras of ##SL(2, \mathbb{C})##, which is simply

    [tex][X_i, X_j] = \varepsilon_{ijk} X_k,[/tex]
    except that we allow complex linear combinations of the generators.
     
  4. Jul 2, 2013 #3
    would it be possible to talk a bit more about the second part, the compactness or non-compactness of a Lie group?

    In any case thank you very much.
     
  5. Jul 2, 2013 #4

    dextercioby

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    Ben nicely touched the compactness of a Lie group from the compactness of its Lie algebra. But a group is a topological space on its own, so its compactness is defined in terms of open sets, open covers and subcovers.
     
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