  We study the discretization of 3d gravity with $\Lambda=0$ following the loop
quantum gravity framework. In the process, we realize that different choices of
polarization are possible. This allows to introduce a new discretization based
on the triad as opposed to the connection as in the standard loop quantum
gravity framework. We also identify the classical non-trivial symmetries of
discrete gravity, namely the Drinfeld double, given in terms of momentum maps.
Another choice of polarization is given by the Chern-Simons formulation of
gravity. Our framework also provides a new discretization scheme of
Chern-Simons, which keeps track of the link between the continuum variables and
the discrete ones. We show how the Poisson bracket we recover between the
Chern-Simons holonomies allows to recover the Goldman bracket. There is also a
transparent link between the discrete Chern-Simons formulation and the
discretization of gravity based on the connection (loop gravity) or triad
variables (dual loop gravity).
