We construct realizations of the generators of the $\kappa$-Minkowski space and $\kappa$-Poincar\'{ e} algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $\kappa$-Poincar\'{ e} algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $\kappa$-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the $\kappa$-Minkowski space.
Categories: math-ph hep-th math.MP