Exploring the Isomorphism between SU(1,1) and SL(2,R)

[Adorno]

Homework Statement

Describe the Lie group SU(1, 1)

Homework Equations

N/A

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.

 

[mighty2000]

Thank you for your post. I can provide some insight into the Lie group SU(1,1). This group is a special unitary group, meaning its elements are complex matrices with determinant equal to 1 and unitary, meaning they satisfy g^{\dagger}g = gg^{\dagger} = I, where I is the identity matrix. In the case of SU(1,1), the matrices are 2x2 and have the additional property of leaving the matrix \eta_2 = \mathrm{diag}(1,-1) invariant.

As you mentioned, the general form of a matrix in SU(1,1) is given by \left( \begin{array}{ccc}
\alpha & \beta \\
\beta^* & \alpha^* \end{array} \right) , where |\alpha|^2 - |\beta|^2 = 1. To parametrize this, we can use the exponential function and hyperbolic trigonometric functions as you suggested. Specifically, we can write the elements of SU(1,1) as \left( \begin{array}{ccc}
\mathrm{cosh}(\theta) & -\mathrm{sinh}(\theta) \\
-\mathrm{sinh}(\theta) & \mathrm{cosh}(\theta) \end{array} \right) , where \theta \in \mathbb{R}. This parametrization satisfies the conditions for the elements of SU(1,1) and allows us to see the group as a hyperbolic rotation in the complex plane.

To show that SU(1,1) is isomorphic to SL(2,R), we can use the Cayley transform, which is a mapping from SU(1,1) to SL(2,R) defined by g \mapsto (I - g)(I + g)^{-1}. This mapping is a homomorphism, meaning it preserves the group structure, and is bijective, meaning it is both injective and surjective. Therefore, it is an isomorphism between the two groups.

I hope this helps you better understand the Lie group SU(1,1) and its relationship to SL(2,R). Keep up the good work in your studies!