\Pi_{\tau}^{-1})$ factorized by a suitable (two-sided) ideal of relations –. Coalgebraic structure $$\begin{gathered}
\Delta_{\tau}(\Pi_{\tau})=\Pi_{\tau}\otimes \Pi_{\tau},
\qquad
\Delta_{\tau}\big(\Pi_{\tau}^{-1}\big)=\Pi_{\tau}^{-1}\otimes \Pi_{\tau}^{-1},
\\
\Delta_{\tau}(P_{i})=P_{i}\otimes \Pi_{\tau}+1\otimes P_{i},
\qquad
i,j=1,\ldots,D-1,
\\
\Delta_{\tau}(M_{ij})=M_{ij}\otimes 1+1\otimes M_{ij},
\\
\Delta_{\tau}(M_{\tau i})=M_{\tau i}\otimes 1+\Pi_{\tau}^{-1}\otimes M_{\tau i}+\frac{\tau^{2}}{\kappa}P^{j}\Pi_{\tau}^{-1}\otimes M_{ij},
\label{q3}\end{gathered}$$ as well as antipodes $$\begin{gathered}
S_{\tau}(P_{i})=- P_{i}\Pi_{\tau}^{-1},
\qquad
S_{\tau}(\Pi^{\pm 1}_{\tau})=\Pi^{\mp 1}_{\tau},
\qquad
S_{\tau}(M_{ij})=-M_{ij},
\\
S_{\tau}(M_{\tau i})=-M_{\tau i}-\frac{\tau^{2}}{\kappa}
P^{k}M_{ki}-\frac{\tau^{2}}{2}\left(\Pi_{\tau}-\Pi_{\tau}^{-1}\left(1+ \frac{1}{\kappa^{2}}P^m P_{m}\right)\right)
M_{\tau i}-\frac{\tau^{2}}{2\kappa^{2}}C_{\tau}M_{\tau i}\!\!\!
\label{q4}\end{gathered}$$ can be calculated from –.

In order to complete the definition one leaves counit undeformed, i.e., $\epsilon (X)=0$ for $X=(M_{ij},M_{\tau i},P_{i})$ and $\epsilon (\Pi_{\tau})=1=\epsilon (\Pi_{\tau}^{-1})$. Deformed and undeformed central elements can now be expressed as $$\begin{gathered}
\label{q6}
C_{\tau}=\frac{\kappa^{2}}{\tau^{2}}\left(\Pi_{\tau}+\Pi_{\tau}^{-1}-2+\frac{\tau^{2}}{\kappa^{2}}P^m
P_{m}\Pi_{\tau}^{-1}\right),
\qquad
C=C_\tau\left(1+\frac{\tau^2}{4\kappa^2}C_\tau\right).\end{gathered}$$ It is important to note that the Hopf algebra $U_{q,\tau}(\operatorname{iso}(g))$ can be considered as an algebra over polynomial ring $\mathbb{C}[{1\over\kappa}]$ (instead of $h$-adic ring $\mathbb{C}[[{1\over \kappa}]]$), since it contains only polynomial expressions in the formal variable ${\frac{1}{\kappa}}$. Because of this we are entitled to introduce a new (non-isomorphic) Hopf algebra over $\mathbb{C}$ by assigning some numerical (complex in general) value to the parameter $\kappa$ in the formulas –. A real value of $\kappa$ is necessary if one wants to preserve the real form  of the Hopf algebra. This solves the so-called specialization problem for $U_{q,\tau}(\operatorname{iso}(g))$ in the case 