  We construct a certain cross product of two copies of the braided dual
$\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic
double $E_H$, and which we use to construct representations of the punctured
elliptic braid group extending the well-known representations of the planar
braid group attached to $H$. We show that the elliptic double is the universal
source of such representations. We recover the representations of the punctured
torus braid group obtained in arXiv:0805.2766, and hence construct a
homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is
factorizable.
  The universal property of $E_H$ endows it with an action by algebra
automorphisms of the mapping class group $\widetilde{SL_2(\mathbb{Z})}$ of the
punctured torus. One such automorphism we call the quantum Fourier transform;
we show that when $H=U_q(\mathfrak{g})$, the quantum Fourier transform
degenerates to the classical Fourier transform on $D(\mathfrak{g})$ as $q\to
1$.
