isomorphism of the group PSL_2(C) with the group of Möbius transformations

We identify the group $G$ of Möbius transformations with the projective special linear group $PSL_{2}(\mathbb{C})$ . The isomorphism $\Psi$ (of topological groups) is given by $\Psi:\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\mapsto{\frac{az+b}{cz+d}}$ . (Here, the notation $[M]$ means the equivalence class $[M]=\{Mt\mid t\in\mathbb{C}\}$ )

This mapping is:

Well-defined:

If $\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]=\left[{\left(\begin{smallmatrix}{a^{% \prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]$ then $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})=t(a,b,c,d)$ for some $t$ , so $z\mapsto{\frac{az+b}{cz+d}}$ is the same transformation as $z\mapsto{\frac{a^{\prime}z+b^{\prime}}{c^{\prime}z+d^{\prime}}}$ .

A homomorphism:

Calculating the composition

\left.{\frac{az+b}{cz+d}}\right|_{z={\frac{ew+f}{gw+h}}}=\frac{a{\frac{ew+f}{% gw+h}}+b}{c{\frac{ew+f}{gw+h}}+d}=\frac{(ae+bg)w+(af+bh)}{(ce+dg)w+(cf+dh)}

we see that $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)\cdot\Psi\left(\left[{\left(% \begin{smallmatrix}{e}&{f}\\ {g}&{h}\end{smallmatrix}\right)}\right]\right)=\Psi\left(\left[{\left(\begin{% smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\cdot\left[{\left(\begin{smallmatrix}{e% }&{f}\\ {g}&{h}\end{smallmatrix}\right)}\right]\right)$ .

A monomorphism:

If $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)=\Psi\left(\left[{\left(\begin{% smallmatrix}{a^{\prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]\right)$ , then it follows that $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})=t(a,b,c,d)$ , so that $\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]=\left[{\left(\begin{smallmatrix}{a^{% \prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]$ .

An epimorphism:

Any Möbius transformation $z\mapsto{\frac{az+b}{cz+d}}$ is the image $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)$ .

Title	isomorphism of the group PSL_2(C) with the group of Möbius transformations
Canonical name	IsomorphismOfTheGroupPSL2CWithTheGroupOfMobiusTransformations
Date of creation	2013-03-22 12:43:30
Last modified on	2013-03-22 12:43:30
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Result
Classification	msc 57S25