and because of the homomorphism property, also for the entire algebra $H$. The first equalities in Eqs.- were calculated by using $e^{\rho}he^{-\rho} =e^{ad\rho}h$, and in the second equality of Eqs. and relations $\widetilde{\mathcal{R}}$ (, ) were used.\
In [@Young:2008zm] the authors have used a perturbative approach to construct the universal $R$-matrix for the $\kappa$-deformed Poincaré algebra generated by $p_\mu$ and $\hat{M}_{\mu \nu}$ from the standard basis of [@Luk1] (case (ii) in Section \[IV\]) up to terms of order $\kappa^{-5}$ ($a_0=-1/\kappa$). The $R$-matrix they presented is expanded in terms of the deformed Poincaré generators and possesses a wedge-product structure. They give no statement about the existence or appearance of the matrix at higher orders of the deformation parameter. We have shown here that the exact $R$-matrix is precisely given by .

Our method for expanding $R=e^{\rho}, \, (\rho= i(A\otimes x_kp_k
-x_kp_k\otimes A))$ in terms of $p_\mu$ and $\hat{M}_{\mu\nu}$ is the following. We wish to find $r_1,r_2, ...$ such that $R=\exp(r_1+r_2+r_3+...)$, where $r_i$ is a series in $a_0,$ starting with $a_0^{i}$ and is linear in $\hat{M}_{\mu\nu}$ (we do not consider terms containing the momenta alone), for all $\, i=1,2,3,...$. It follows that $\rho +\frac{1}{2}\rho^{2}+\frac{1}{6}\rho^3+...=r_1+r_2+\frac{1}{2}r_1^{2} +\frac{1}{2}(r_1r_2+r_2r_1) +r_3+ \frac{1}{6}r_1^3+...$.

In the first order we 