I'm trying to work out some differential geometry of the matrix Lie group $SU(1,1)$. It is the group of $2\times 2$ complex matrices such that:
$$ (U^{*})^{T}\eta U=\eta $$ where:
$$ \eta=\left(\begin{array}{cc} 1 & 0 \\ 0 &-1\end{array}\right) $$ From this it follows:
$$ U=\left(\begin{array}{cc}\alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{array}\right) $$ and:
$$ \alpha\overline{\alpha} - \beta\overline{\beta}=1 $$ This means there are three free parameters, one possible choice is $(\omega,\gamma,\theta)$ such that:
$$ U=\left(\begin{array}{cc} \exp(\imath\omega)\cosh(\theta) & \exp(\imath\gamma)\sinh(\theta) \\ \exp(-\imath\gamma)\sinh(\theta) & \exp(-\imath\omega)\cosh(\theta) \end{array}\right) $$ Where $U$ is the image of $g\in SU(1,1)$ through its embedding $i$ in $GL_{2}(\mathbb{C})$. Now the coordinates $(\omega,\gamma,\theta)$ do not allow me to find the elements of a basis in $T_{e}SU(1,1)$ as derivatives of the coordinate curves because the coordinate curve of $\gamma$ does not pass on the identity element $e$. I thus tried another way and found that the defining condition of $SU(1,1)$ and the properties of the exponential map allow to write the Lie algebra $\mathfrak{su}(1,1)$ as:
$$ A: (A^{*})^{T}\eta -\eta A= \;\;\;Tr(A)=0 $$ and thus to find three matrices:
$$ A_{1}=\left(\begin{array}{cc} 0 & \imath \\ -\imath & 0\end{array}\right) $$ $$A_{2}=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right) $$ $$ A_{3}=\left(\begin{array}{cc} \imath & 0 \\ 0 &-\imath\end{array}\right) $$
It turns out that:
$$ A_{2}=(i_{*})_{e}\frac{d c_{2}(t)}{dt}\left|_{t=0}\right.\;\;\;c_{2}(t)=\left\{\begin{array}{l}c_{2}^{\omega}(t)=0\\c_{2}^{\theta}(t)=t\\c_{2}^{\gamma}(t)=0\end{array}\right. $$
$$ A_{3}=(i_{*})_{e}\frac{d c_{3}(t)}{dt}\left|_{t=0}\right.\;\;\;c_{3}(t)=\left\{\begin{array}{l}c_{3}^{\omega}(t)=t\\c_{3}^{\theta}(t)=0\\c_{3}^{\gamma}(t)=0\end{array}\right. $$
My question is: what curve $c_{1}(t)$ through $e$ is such that its derivative at $e$ is $A_{1}=(i_{*})_{e}\frac{dc_{1}(t)}{dt}\left|_{t=0}\right.$?
I need to know it because I'm interested in the left invariant vector fields on $SU(1,1)$.
I hope someone can help me. Thank you.
I believe $$\gamma(t)= \left( \begin{array}{cc} 2e^{-iu}-e^{-2iu} & \sqrt{2}e^{2iu}-\sqrt{2}e^{iu} \\ \sqrt{2}e^{-2iu}-\sqrt{2}e^{-iu} & 2e^{iu}-e^{2iu} \\ \end{array} \right)$$ will work. Here $u=t/\sqrt{2}$