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We show how the constant curvature spacetimes of 3d gravity and theassociated symmetry algebras can be derived from a single quantum deformationof the 3d Lorentz algebra sl(2,R). We investigate the classical Drinfel'ddouble of a "hybrid" deformation of sl(2,R) that depends on two parameters(\eta,z). With an appropriate choice of basis and real structure, thisDrinfel'd double agrees with the 3d anti-de Sitter algebra. The deformationparameter \eta is related to the cosmological constant, while z is identifiedwith 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 3dhyperbolic space through analytic continuation in \eta and z; we alsoinvestigate the limits of vanishing \eta and z, which yield the flat spacetimes(Minkowski and Euclidean spaces) and Newtonian models, respectively.

Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations