Dynamical emergence of SUq(2) from the regularization of (2+1)D gravity with a cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in (2 + 1)D has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an SUq(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 + 1)D gravity with a nonzero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the SUq(2) quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of SUq(2) that are recovered for any Wilson loop evaluated at the SU(2) group element that encodes the discretization of the spacetime 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 SUq(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 we derive explicitly the partition function amplitude of the Turaev-Viro model.