Classically the group of Moebius transformations of the unit disk and Moebius transformations of the upper half plane are isomorphic, as the unit disk and upper half plane are transformed into each other by another Moebius transformation. We can ask whether the isomorphism of groups works in the noncommutative case, even if the underlying motivation may not work. The quantum groups $C_q[SU_{1,1}]$ and $C_q[SL_{2}(R)]$ are the same as Hopf algebras (the same as $C_q[SU_{2}]$), but they have a different star structure: $$ C_q[SU_{1,1}]:\quad q^*=q,\quad \begin{pmatrix}a&b\\ c&d\end{pmatrix}^*=\begin{pmatrix}d&q^{-1}c\\ qb&a\end{pmatrix}$$ $$ C_q[SL_{2}(R)]:\quad q^*=q^{-1},\quad \begin{pmatrix}a&b\\ c&d\end{pmatrix}^*=\begin{pmatrix}a& q^{-1} b\\ q c&d\end{pmatrix}$$ Well, there is another complication - the $q$s have to change, as one is real and the other is on the unit circle, so we really ask if there is an isomorphism $C_q[SU_{1,1}] \cong C_p[SL_{2}(R)]$ for $p$ a function of $q$. Also the determinants may vary - strictly the Mobeius groups are projective $PSL_2(R)$, or alternatively we might work without the determinant=1 condition on $GL_2(R)$ - whatever is needed to get an interesting noncommutative result.
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