the theory of realizations [@ms06],[@starproductrealisations],[@km11] the Lorentz generators can be expressed in terms of $x$ and $ p$ generators of the Heisenberg algebra. To make the picture closed, let us mention that in the theory of realizations, there is one realization corresponding to each twist element. More concretely, having a twist element $\mathcal{F},$ the corresponding realization can be obtained as $$\label{twistrealization}
  \hat{x}_\mu = m_0 \bigg( \mathcal{F}^{-1} \triangleright (x_\mu \otimes id )  \bigg),$$ where $m_0$ is the multiplication map. The family of twists leads to $$\label{hxl}
\hat{x}_i= x_i Z^{-\lambda}, \quad \hat{x}_0=x_0 -a_0(1-\lambda)x_kp_k.$$ One can check that realizations for $\hat{x}_\mu$ satisfy the commutation relations $$\left[\hat{x}_\mu,\hat{x}_\nu\right] =i\left(a_\mu\hat{x}_\nu-a_\nu\hat{x}_\mu
  \right),
  \label{kappa}$$ that are known to describe $\kappa$-Minkowski spacetime algebra. In this way the $\kappa$-deformation of the Heisenberg algebra $H$ is made explicit.

Eq., along with -, can be used to derive $\Delta \hat{x}_\mu$ $$\Delta \hat{x}_i = \Delta(x_iZ^{-\lambda}) = (\Delta x_i) (\Delta Z^{-\lambda}) = \left\{ \begin{array}{clrr} & (x_i\otimes Z^{\lambda}) (Z^{-\lambda}\otimes Z^{-\lambda}) = x_iZ^{-\lambda}\otimes 1= \hat{x}_i\otimes 1 \\
& (Z^{\lambda-1}\otimes x_i) (Z^{-\lambda}\otimes Z^{-\lambda}) = Z^{-1}\otimes x_iZ^{-\lambda}= Z^{-1} \otimes \hat{x}_i,
\end{array} \right .$$ and analogously for $\hat{x}_0$, with the result $$\Delta \hat{x}_0= \hat{x}_0\otimes 1 = 1\otimes \hat{x}_0 -a_0p_kZ^{\lambda-1}\otimes \hat{x}_k .$$ These results can be written compactly as $$\Delta 