  The quantization of the reduced phase-space of the Einstein-Hilbert action
for gravity in $2+1D$ has been shown to bring about the emergence, at the
quantum level, of a topological quantum field theory endowed with an $SU_q(2)$
quantum group symmetry structure. We hereby tackle the same problem, but start
from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$
gravity with non-zero cosmological constant in the Palatini formalism, and
subsequently impose the constraints. We hence show the dynamical emergence of
the $SU_q(2)$ quantum group at the quantum level within the spin-foam
framework. The regularized curvature constraint is responsible for the
effective representations of $SU_q(2)$ that are recovered for any Wilson loop
evaluated at the $SU(2)$ group element that encodes the discretization of the
space-time curvature induced by the cosmological constant. The extension to the
spin-network basis, and consequently to any transition amplitude between its
generic states, enables us to derive in full generality the recoupling theory
of $SU_q(2)$. We provide constructive examples for the scalar product of two
loop states and spin-networks encoding trivalent vertices. We further comment
on the diffeomorphism symmetry generated by the implementation of the curvature
constraint, and finally derive explicitly the partition function amplitude of
the Turaev-Viro model.

General Relativity and Quantum Cosmology;High Energy Physics - Theory