  In this note, we propose a decomposition of the quantum matrix group
SL$_q^+(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebra
generators of Faddeev's modular double of $\text{U}_q(\mathfrak{sl}(2,
\mathbb{R}))$. The formula is checked by relating hyperbolic representation
matrices with the Whittaker function. We interpret (or derive) it in terms of
Hopf duality, and use it to explicitly construct the regular representation of
the modular double, leading to the Casimir and its modular dual as the analogue
of the Laplacian on the quantum group manifold. This description is important
for both 2d Liouville gravity, and 3d pure gravity, since both are governed by
this algebraic structure. This result builds towards a $q$-BF formulation of
the amplitudes of both of these gravitational models.
