  We show how the constant curvature spacetimes of 3d gravity and the
associated symmetry algebras can be derived from a single quantum deformation
of the 3d Lorentz algebra sl(2,R). We investigate the classical Drinfel'd
double of a "hybrid" deformation of sl(2,R) that depends on two parameters
(\eta,z). With an appropriate choice of basis and real structure, this
Drinfel'd double agrees with the 3d anti-de Sitter algebra. The deformation
parameter \eta is related to the cosmological constant, while z is identified
with the inverse of the speed of light and defines the signature of the metric.
We generalise this result to de Sitter space, the three-sphere and 3d
hyperbolic space through analytic continuation in \eta and z; we also
investigate the limits of vanishing \eta and z, which yield the flat spacetimes
(Minkowski and Euclidean spaces) and Newtonian models, respectively.
