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A classical deformation procedure, based on universal enveloping algebras,Casimirs and curvatures of symmetrical homogeneous spaces, is applied toseveral cases of physical relevance. Starting from the (3+1)D Galilei algebra,we describe at the level of representations the process leading to its twophysically meaningful deformed neighbours. The Poincare algebra is obtained byintroducing a negative curvature in the flat Galilean phase space (or space ofworldlines), while keeping a flat spacetime. To be precise, starting from arepresentation of the Galilei algebra with both Casimirs different from zero,we obtain a representation of the Poincare algebra with both Casimirsnecessarily equal to zero. The Poincare angular momentum, Pauli-Lubanskicomponents, position and velocity operators, etc. are expressed in terms of"Galilean" operators through some expressions deforming the proper Galileanones. Similarly, the Newton-Hooke algebras appear by endowing spacetime with anon-zero curvature, while keeping a flat phase space. The same approach,starting from the (3+1)D Poincare algebra provides representations of the(anti)de Sitter as Poincare deformations.Comment: 19 pages, LaTeX.7. Comments and references adde

(Anti)de Sitter/Poincaré symmetries and representations from Poincaré/Galilei through a classical deformation approach