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Description of SU(1, 1)

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data
    Describe the Lie group SU(1, 1)

    2. Relevant equations
    N/A

    3. The attempt at a solution
    [itex]\mathrm{SU}(1, 1)[/itex] is the group of all non-singular 2 x 2 matrices which leave the matrix [itex]\eta_2 = \mathrm{diag}(1, -1)[/itex] invariant, i.e. all [itex]g \in M_{2 \times 2}(\mathbb{C})[/itex] such that [itex]g^{\dagger}\eta_2g = \eta_2[/itex]. I know that the general form of a matrix in [itex]\mathrm{SU}(1, 1)[/itex] is given by [itex]\left( \begin{array}{ccc}
    \alpha & \beta \\
    \beta^* & \alpha^* \end{array} \right) [/itex], where [itex]|\alpha|^2 - |\beta|^2 = 1[/itex]. However, I think I need to "parametrise" the entries in the matrix somehow. For example, the elements of the group [itex]\mathrm{SO}(2)[/itex] have the form [itex]\left( \begin{array}{ccc}
    a & -b \\
    b & a \end{array} \right) [/itex], where [itex]a[/itex], [itex]b \in \mathbb{R}[/itex] and [itex]a^2 + b^2 = 1[/itex]. These can be parametrised by [itex]\left( \begin{array}{ccc}
    \mathrm{cos}(\theta) & -\mathrm{sin}(\theta) \\
    \mathrm{sin}(\theta) & \mathrm{cos}(\theta) \end{array} \right) [/itex], for [itex]\theta \in (-\pi, \pi] [/itex]. I need to get a similar kind of parametrisation for the elements in [itex]\mathrm{SU}(1, 1)[/itex]. I'm pretty sure that such a parametrisation would involve 2 or 3 parameters, as well as the exponential function and hyperbolic trigonometric functions. I can't quite see how to get it based on the intrinsic form of the group elements though.

    Also, I need to show that this group is isomorphic to the group [itex]\mathrm{SL}(2, \mathbb{R})[/itex], and I'm not entirely sure how to do this. I've read somewhere that the "Cayley transform" gives an isomorphism, but I don't really know what that is. Any help would be appreciated.
     
  2. jcsd
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