  It is well-known that quantum groups are relevant to describe the quantum
regime of 3d gravity. They encode a deformation of the gauge symmetries
parametrized by the value of the cosmological constant. They appear as a form
of regularization either through the quantization of the Chern-Simons
formulation or the state sum approach of Turaev-Viro. Such deformations are
perplexing from a continuum and classical picture since the action is defined
in terms of undeformed gauge invariance. We present here a novel way to derive
from first principle and from the classical action such quantum group
deformation. The argument relies on two main steps. First we perform a
canonical transformation, which deformed the gauge invariance and the boundary
symmetries, and makes them depend on the cosmological constant. Second we
implement a discretization procedure relying on a truncation of the degrees of
freedom from the continuum.
