  We use factorisations of the local isometry groups arising in 3d gravity for
Lorentzian and Euclidean signatures and any value of the cosmological constant
to construct associated bicrossproduct quantum groups via semidualisation. In
this way we obtain quantum doubles of the Lorentz and rotation groups in 3d, as
well as kappa-Poincare algebras whose associated r-matrices have spacelike,
timelike and lightlike deformation parameters. We confirm and elaborate the
interpretation of semiduality proposed in [13] as the exchange of the
cosmological length scale and the Planck mass in the context of 3d quantum
gravity. In particular, semiduality gives a simple understanding of why the
quantum double of the Lorentz group and the kappa-Poincare algebra with
spacelike deformation parameter are both associated to 3d gravity with
vanishing cosmological constant, while the kappa-Poincare algebra with a
timelike deformation parameter can only be associated to 3d gravity if one
takes the Planck mass to be imaginary.
