Deformed Maxwell Algebras and their Realizations

We study all possible deformations of the Maxwell algebra. In D = d + 1 not equal 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d + 1, 1) circle plus so(d, 1) or to so(d, 2) circle plus so(d, 1) depending on the signs of the deformation parameter. We construct in the dS(AdS) space a model of massive particle interacting with Abelian vector field via non-local Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We construct a phase diagram, with two parts of the (b, k) plane with so(3, 1) circle plus so(2, 1) and so(2, 2) circle plus so(2, 1) algebras separated by a critical curve along which the algebra is isomorphic to Iso(2, 1) circle plus so(2, 1). We introduce in D=2+1 the Volkov-Akulov type model for a Abelian Goldstone-Nambu vector field described by a non-linear action containing as its bilinear term the free Chern-Simons Lagrangean.