A group theoretic description of the $\kappa$-Poincar\'e Hopf algebra

It is well known in the literature that the momentum space associated to the $\kappa$-Poincar'e algebra is described by the Lie group $\mathsf{A}\mathsf{N}(3)$. In this letter we show that the full $\kappa$-Poincar'e Hopf algebra structure can be obtained from rather straightforward group-theoretic manipulations starting from the Iwasawa decomposition of the of the $\mathsf{S}\mathsf{O}(\mathsf{1},\mathsf{4})$ group.