Exercise 3, § 3

Let E be the complex vector space of symmetric complex matrices with 2 rows and 2 columns. For all $s \in \mathbf{SL}(2, \mathbf{C})$, we define an automorphism $\rho(s)$ of E by $\rho(s).m = s.m.s^t$. (a) Show that $\rho$ is an analytic linear representation of $\mathbf{SL}(2, \mathbf{C})$. Identifying $\begin{pmatrix} a & b \\ b & c \end{pmatrix} \in E$ with $(a, b, c) \in \mathbf{C}^3$, the matrix of $\rho \left( \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \right)$ is

(b) Show that $m \mapsto \det(m)$ is the non-degenerate quadratic form on E and that $\rho$ is a morphism of $\mathbf{SL}(2, \mathbf{C})$ onto $\mathbf{SO}(q)$ with kernel $\{I, -I\}$. (For the surjectivity, observe that, in the matrix of an element of $\mathbf{SO}(q)$, the 1st column can be expressed in the form $(\alpha^2, \alpha\gamma, \gamma^2)$ and the 3rd column in the form $(\beta^2, \beta\delta, \delta^2)$, where $\alpha^2\delta^2 + \beta^2\gamma^2 - 2\alpha\gamma\beta\delta = 1$. Then observe that 2 elements of $\mathbf{SO}(q)$ which coincide on a non-isotropic plane are equal.) (c) Deduce that the complex Lie group $\mathbf{SO}(3, \mathbf{C})$ is isomorphic to the complex Lie group $\mathbf{SL}(2, \mathbf{C})/\{I, -I\}$.