# Description of SU(1, 1)

1. Aug 25, 2011

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
$\mathrm{SU}(1, 1)$ is the group of all non-singular 2 x 2 matrices which leave the matrix $\eta_2 = \mathrm{diag}(1, -1)$ invariant, i.e. all $g \in M_{2 \times 2}(\mathbb{C})$ such that $g^{\dagger}\eta_2g = \eta_2$. I know that the general form of a matrix in $\mathrm{SU}(1, 1)$ is given by $\left( \begin{array}{ccc} \alpha & \beta \\ \beta^* & \alpha^* \end{array} \right)$, where $|\alpha|^2 - |\beta|^2 = 1$. However, I think I need to "parametrise" the entries in the matrix somehow. For example, the elements of the group $\mathrm{SO}(2)$ have the form $\left( \begin{array}{ccc} a & -b \\ b & a \end{array} \right)$, where $a$, $b \in \mathbb{R}$ and $a^2 + b^2 = 1$. These can be parametrised by $\left( \begin{array}{ccc} \mathrm{cos}(\theta) & -\mathrm{sin}(\theta) \\ \mathrm{sin}(\theta) & \mathrm{cos}(\theta) \end{array} \right)$, for $\theta \in (-\pi, \pi]$. I need to get a similar kind of parametrisation for the elements in $\mathrm{SU}(1, 1)$. 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 $\mathrm{SL}(2, \mathbb{R})$, 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.