  We study the quantization of the corner symmetry algebra of 3d gravity
associated with 1d spatial boundaries. We first recall that in the continuum,
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 with a
$\mathcal{D}\mathrm{SU}(2)$ quantum symmetry group that depends on an integer
$N$. 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 a path integral amplitudes for 3d loop quantum gravity. We then
show that we recover in the $N\rightarrow\infty$ limit the central extension of
the Poincar\'e current algebra. The number of boundary edges defines a
discreteness parameter $N$ which counts the number of flux lines attached to
the boundary. We analyse the refinement, coarse-graining and fusion processes
as $N$ changes. Identifying a discrete current algebra on quantum boundaries is
an important step towards understanding how conformal field theories arise on
spatial boundaries in loop quantum gravity. It also shows how an asymptotic BMS
symmetry group could appear from the continuum limit of 3d quantum gravity.
