Special Relativity as a non commutative geometry: Lessons for Deformed Special Relativity

24 pages, RevTeX4Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is there to tell us what is the underlying physics, in such a way that we can understand and interpret these ambiguities. We use these insights to comment on the physics of DSR