Constructing an Analytic Mapping for SL(2;R) using Quadratic Forms

[r16]

Homework Statement

Construct the analytic mapping \phi(x,y) for the H^{2+} \times S^1 representation of SL(2;R)

Homework Equations

g(x) \circ g(y) = g(\phi(x,y))

The Attempt at a Solution

So, all points in SL(2;R) lie on the manifold H^{2+} \times S^1. I also know that SL(2;R) is 3 dimensional, so I will parametrize it as x=[x y \theta].

For a point to lie on H^{2+} it has to satisfy the quadratic form z^2 -x^2 -y^2=1 in R^3. I calculated the 3x3 matrix, H, for this quadratic form, which has diagonal [-1 -1 1] and zeros everywhere else.

My goal is to calculate the matrix rep for g(x) \in SL(2;R) by multiplying H and the rotation matrix for S^1 which is well known, and then using this information end up solving for \phi

My problem is that H, my matrix for the quadratic form, is not paramaterized by x and y, its elements are just constants. How do I find a quadratic form for H^2 that is paramaterized by x and y?

 

[arkajad]

The mapping \phi is supposed to be from where to where? Real analytic or complex analytic?