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Homework Statement
Construct the analytic mapping [tex]\phi(x,y)[/tex] for the [tex]H^{2+} \times S^1[/tex] representation of [tex]SL(2;R)[/tex]
Homework Equations
[tex]g(x) \circ g(y) = g(\phi(x,y))[/tex]
The Attempt at a Solution
So, all points in SL(2;R) lie on the manifold [tex]H^{2+} \times S^1[/tex]. I also know that SL(2;R) is 3 dimensional, so I will parametrize it as x=[x y [tex]\theta[/tex]].
For a point to lie on [tex]H^{2+}[/tex] it has to satisfy the quadratic form [tex]z^2 -x^2 -y^2=1[/tex] in [tex]R^3[/tex]. 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 [tex]g(x) \in SL(2;R) [/tex] by multiplying H and the rotation matrix for [tex]S^1[/tex] which is well known, and then using this information end up solving for [tex]\phi[/tex]
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 [tex]H^2[/tex] that is paramaterized by x and y?