Special relativity as a noncommutative geometry: Lessons for deformed special relativity

Deformed special relativity (DSR) is obtained by imposing a maximal energy to special relativity and deforming the Lorentz symmetry (more exactly, the Poincar\'e symmetry) to accommodate this requirement. One can apply the same procedure in the context of Galilean relativity by imposing a maximal speed (the speed of light). Effectively, one deforms the Galilean group and this leads to a noncommutative space structure, together with the 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 we can understand and interpret these ambiguities. We use these insights to comment on the physics of DSR.