Deformed Maxwell Algebras and their Realizations

We study all possible deformations of the Maxwell algebra. In D = d +1 6= 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d + 1,1)⊕ so(d,1) or to so(d,2)⊕ 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 nonlocal 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)⊕ so(2,1) and so(2,2)⊕ so(2,1) algebras separated by a critical curve along which the algebra is isomorphic to Iso(2,1)⊕ 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.