Linear representation of special euclidean group as subgroup of GL2(C)

Question

I came across this question:

Find an isomorphism from the group of orientation preserving isometries of the plane to some subgroup of $GL_{2}(\mathbb C)$.

I'm having trouble with finding such isomorphism. Mainly, I'm having trouble with finding some representation of those isometries in a way that would respect the group structure. I know that you can represent any isometry of the plane using a $3\times 3$ real matrix, and I want to somehow use this representation in order to construct a new one (maybe since I'm looking at a smaller group and can also use complex numbers I can somehow use a smaller matrix), but I'm not entirely sure on how to do this.

Any help would be appreciated - I'd like to know how one can approach such a problem.

Thanks in advance

Answer 1

We can represent any orientation preserving isometry with a single angle of rotation and a single vector of translation (or simply a complex number). We can thus use the following monomorphism into $GL(2,\mathbb C)$: $$(\alpha,x+iy)\mapsto\begin{bmatrix} e^{i\alpha}&x+iy\\0&1\end{bmatrix}$$

Note that this representation preserves the group structure, and it acts on the vector $\begin{bmatrix} z \\ 1\end{bmatrix}$ by sending it to $\begin{bmatrix} ze^{i\alpha}+x+iy \\ 1\end{bmatrix}$, which is (in the first component) exactly the image of $z$ under our given isometry.

Answer 2

Hint: Think of affine complex linear-fractional transformations of the form $z\mapsto az+b$, $|a|=1$, and the epimorphism $GL(2, {\mathbb C})\to PGL(2, {\mathbb C})$, the group of complex linear-fractional transformations.

Edit: the epimorphism is standard: $$\left[ \begin{array}{cc} a&b\\ c&d\\ \end{array}\right]\mapsto \frac{az +b}{cz+d} $$