Drinfel'd double structures for Poincare and Euclidean groups

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel'd double structure for this group is explicitly given. The similar construction for the Poincare case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel'd double structures of the so(3, 1) Lie algebra is presented.