Computing the infinitesimal generators for the Mobius transformation

[platypi]

Homework Statement

The Mobius transformation is $$\frac{at+b}{ct+b},$$ with the constraint $ad-bc=1$. Find the infinitesimal generators of its Lie algebra.

Relevant Equations

N/A

I don't know where to start. I understand that the constraint $ad-bc=1$ gives us one less parameter since $d=1+bc/a$. So we can rewrite our original function. I know how to compute the generators of matrix groups but in this case the generators will be functions. I also know there should be three of them since we have three independent parameters. However, I'm not sure what to do. I think we may have to take partial derivatives?

 

[strangerep]

[ I presume your denominator should be $ct+d$ ? ]

Suppose you have a transformation of the form $$ t' = t'(t,a,b,c,...)~,$$where $a,b,c$ are parameters of the transformation.

The general formula for the generator of such a coordinate transformation is $$X_a ~=~ \left. \frac{\partial t'}{\partial a} \right|_{a,b,c=\text{Id}} \; \frac{\partial}{\partial t}$$and similarly for $b,c,...$

The "Id" notation denotes whatever value the parameter has at the identity transformation. E.g., in the Mobius case, for $t' = t$ we must have $a=d=1$, and $b = c = 0$.