Realizations of

Realizations of $$\kappa $$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the $$\mathfrak {gl}(n)$$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. $$\kappa $$-Deformed $$\mathfrak {igl}(n)$$-Hopf algebras are presented for all cases. The $$\kappa $$-Poincaré–Weyl and $$\kappa $$-Poincaré–Hopf algebras are discussed. The left–right dual $$\kappa $$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to $$\kappa $$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed