  We study the quantization of the corner symmetry algebra of 3d gravity, that
is the algebra of observables associated with 1d spatial boundaries. In the
continuum field theory, at the classical level, this symmetry algebra is given
by the central extension of the Poincar\'e loop algebra. At the quantum level,
we construct a discrete current algebra based on a quantum symmetry group given
by the Drinfeld double $\mathcal{D}\mathrm{SU}(2)$. Those discrete currents
depend on an integer $N$, a discreteness parameter, understood as the number of
quanta of geometry on the 1d boundary: low $N$ is the deep quantum regime,
while large $N$ should lead back to a continuum picture. We show that this
algebra satisfies two fundamental properties. First, it is compatible with the
quantum space-time picture given by the Ponzano-Regge state-sum model, which
provides discrete path integral amplitudes for 3d quantum gravity. The integer
$N$ then counts the flux lines attached to the boundary. Second, we analyse the
refinement, coarse-graining and fusion processes as $N$ changes, and we show
that the $N\rightarrow\infty$ limit is a classical limit where we recover the
Poincar\'e current algebra. Identifying such a discrete current algebra on
quantum boundaries is an important step towards understanding how conformal
field theories arise on spatial boundaries in quantized space-times such as in
loop quantum gravity.
