=p_\mu x_\nu \otimes 1+ x_\nu \otimes p_\mu =                          p_\mu x_\nu \otimes 1+1 \otimes x_\nu p_\mu \\
& (p_\mu \otimes 1+1 \otimes p_\mu) (1 \otimes x_\nu ) =p_\mu \otimes x_\nu +1\otimes p_\mu x_\nu = x_\nu p_\mu \otimes 1+1 \otimes p_\mu x_\nu
\end{array} \right .\end{aligned}$$ In complete analogy, one also obtains four expressions for $\Delta_0(x_\nu p_\mu)$. It now follows, from these expressions, that $$\Delta_0\Bigl( [p_\mu, x_\nu ]\Bigr) =\Delta_0(p_\mu x_\nu)- \Delta_0(x_\nu p_\mu) =[\Delta_0 p_\mu ,\Delta_0 x_\nu] =-i\eta_{\mu\nu} \cdot 1\otimes 1,$$ showing that the coproduct $\Delta_0 h, \, h\in H,$ is compatible with the relations $\mathcal{R}_0$. Hence, $\Delta_0 (H) = \Delta_0 ({\mathcal{A}})  \Delta_0 (T) $ is the algebra isomorphic to $H$. The coalgebra on the undeformed Heisenberg algebra $H$ is also undeformed. The coalgebra of the Heisenberg algebra $H$ does not lead to a Hopf algebra. It is related to an algebraic construction whose mathematical structure is described in terms of a Hopf algebroid [@lu]. Note that the elements $L_{\mu \nu} =x_\mu p_\nu$ generate the $\mathfrak{gl}(4)$ algebra and satisfy $\Delta_0 L_{\mu \nu} = L_{\mu \nu} \otimes 1+1 \otimes L_{\mu \nu}$.

In the following, we interpret $\Delta_0 h$ as the operator $ \Delta_0 h: \Delta_0 (H)
 \longrightarrow \Delta_0 (H), $ defined by $ (\Delta_0 h) 