in [@Wess1; @1110.0944; @Meljanac0702215]) $$\begin{gathered}
\Delta_{\tau}(P_{\mu})=P_{\mu}\otimes \Pi_{\tau}+1\otimes
P_{\mu}-\frac{\tau_{\mu}}{\kappa}P^{\alpha}\Pi_{\tau}^{-1}\otimes
P_{\alpha}-\frac{\tau_{\mu}}{2\kappa^{2}}C_{\tau}\Pi_{\tau}^{-1}\otimes P_{\tau},
\label{copP}
\\
\Delta_{\tau}(M_{\mu \nu})=M_{\mu \nu}\otimes 1+1\otimes M_{\mu
\nu}+\frac{1}{\kappa}P^{\alpha}\Pi_{\tau}^{-1}\otimes (\tau_{\nu}M_{\alpha \mu}-\tau_{\mu}M_{\alpha \nu})
\nonumber
\\
\phantom{\Delta_{\tau}(M_{\mu \nu})=}
 - \frac{1}{2\kappa^{2}}C_{\tau}\Pi_{\tau}^{-1}\otimes (\tau_{\mu}M_{\tau \nu}-\tau_{\nu}M_{\tau \mu}),
\label{copM}\end{gathered}$$ where $M_{\tau \lambda}=\tau^{\alpha}M_{\alpha \lambda}$ and $\tau_{\mu}=g_{\alpha \mu}\tau^{\mu}$ denote covariant components of $\tau^{\mu}$ with respect to the metric $g_{\mu \nu}$. In order to complete the Hopf algebra structure one defines counits: $\epsilon (1)=1$, $\epsilon (P_{\mu})=-P_{\mu}$, $\epsilon (M_{\mu \nu})=-M_{\mu \nu}$ and antipodes $$\begin{gathered}
S_{\tau} (P_{\mu} )
=-\left(P_{\mu}+\frac{\tau_{\mu}}{\kappa}\left(C+\frac{1}{2\kappa}P_{\tau}C_{\tau}\right)\right) \Pi_{\tau}^{-1},
\qquad
S_{\kappa}(\Pi_{\tau})=\Pi_{\tau}^{-1},
\label{SP}
\\
S_{\tau} (M_{\mu \nu} )=-M_{\mu \nu}+\frac{1}{\kappa} P^{\alpha} (\tau_{\nu}M_{\alpha
\mu}-\tau_{\mu}M_{\alpha \nu} ) +\frac{1}{2\kappa^{2}}C_{\tau} (\tau_{\nu}M_{\tau \mu}-\tau_{\mu}M_{\tau
\nu} ).
\label{SM}\end{gathered}$$ So defined Hopf algebra structure will be denoted as $U_{\kappa,\tau}(\operatorname{iso}(g))$, i.e., in particular, $U_{\kappa,\tau}(\operatorname{iso}(g))$ $\cong U(\operatorname{iso}(g))[[\frac{1}{\kappa}]]$ as an algebra. It preserves the reality condition (\[RC1\]) induced by the metric signature in the following well-known Hopf algebra reality condition form (see e.g. [@Majid-book]) $$\begin{gathered}
\Delta_{\tau}(X^{\ast})=\Delta_{\tau}(X)^{\ast \otimes \ast}
\qquad
\text{and}
\qquad
S_{\tau}(S_{\tau}(X^{\ast})^{\ast})=X
\label{RC2}\end{gathered}$$ for arbitrary $X\in U_{\kappa,\tau}(\operatorname{iso}(g))$ provided that the vector $\tau^{\mu}$ and the formal parameter ${\frac{1}{\kappa}}$ are real. Above conditions are enough to be checked on self-adjoint generators , . In particular, the identity: $S_{\tau}(S_{\tau}(X))=\Pi_{\tau}^{D-1}X\Pi_{\tau}^{1-D}$ found in [@1110.0944] can be very helpful.

It is important to notice that simultaneous re-scaling of $\tau$ and $\kappa$ 