Kappa-Minkowski space-time and the star product realizations

We investigate a Lie algebra-type $\kappa$-deformed Minkowski space-time with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of $\kappa$-Minkowski space. The coproduct and the star product corresponding to each of them are found. Utilizing the properties of the {\em{natural}} realization, we construct a scalar field theory on $\kappa$-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space.