unital and associative algebra over the complex numbers generated by the elements $(M_{+-},M_{+a},M_{-a},M_{ab},P_{a},P_{-},\Pi_{+},\Pi_{+}^{-1})$ and factorized by the corresponding (two-sided) ideal of above relations.

Then a coalgebra for the $q$-analog version in this case takes the form $$\begin{gathered}
\Delta_{\tau} (P_{a} )=P_{a}\otimes \Pi_{+}+1\otimes P_{a},
\\
\Delta_{\tau} (\Pi_{+} )=\Pi_{+}\otimes \Pi_{+},
\qquad
\Delta_{\tau} \big(\Pi_{+}^{-1} \big)=\Pi_{+}^{-1}\otimes \Pi_{+}^{-1},
\\
\Delta_{\tau}(P_{-})=P_{-}\otimes \Pi_{+}+\Pi_{+}^{-1}\otimes P_{-}-\left(P_{-}+\frac{1}{2\kappa}C\right)\Pi_{+}^{-1}
\otimes  (\Pi_{+}-1 ) -\frac{1}{\kappa}P^{a}\Pi_{+}^{-1}\otimes P_{a},\!\!\!
\\
\Delta_{\tau} (M_{+ -} )=M_{+ -}\otimes 1+\Pi_{+}^{-1}\otimes M_{+ -}-\frac{1}{\kappa}P^{a}\Pi_{+}^{-1}\otimes M_{+ a},
\\
\Delta_{\tau} (M_{- a} )=M_{- a}\!\otimes 1+\Pi_{+}^{-1}\!\otimes M_{- a}-\frac{1}{\kappa}\left(P_{-}\!+\frac{1}{2\kappa}C\right)\Pi_{+}^{-1}\!
\otimes M_{+ a}-\frac{1}{\kappa}P^{b}\Pi_{+}^{-1}\!\otimes M_{ba},\!\!\!
\\
\Delta_{\tau} (M )=M\otimes 1+1\otimes M
\qquad
\text{for}
\quad
M\in \{M_{+ a},M_{ab}\},\end{gathered}$$ where now $C=2\kappa (\Pi_{+}-1) P_{-}+P^{a}P_{a}$. Including the following antipodes $$\begin{gathered}
S_{\tau}(P_{a})=-P_{a}\Pi_{+}^{-1},
\qquad
S_{\tau}(\Pi_{+})=\Pi_{+}^{-1},
\\
S_{\tau}(P_{-})=-P_{-}\Pi_{+}-\frac{1}{2\kappa}(1+\Pi_{+}) P_{a}P^{a}\Pi_{+}^{-1},
\\
S_{\tau}(M_{+-})=-\Pi_{+}M_{+-}-\frac{1}{\kappa} P^{a}M_{+a},
\\
S_{\tau}(M_{-a})=-\Pi_{+}M_{-a}-\frac{1}{\kappa}\left(P_{-}+\frac{1}{2\kappa}C\right)M_{+a}-\frac{1}{\kappa}P^{b}M_{ba},
\\
S_{\tau}(M)=-M
\qquad
\text{for}
\quad
M\in \{M_{+a},M_{ab}\}
\label{Spm_q}\end{gathered}$$ one completes the Hopf algebra structure provided the counits remain undeformed, i.e. $\epsilon(X)=0$ for $X=(M_{+-},M_{+a},M_{-a},M_{ab},P_{a},P_{-},\Pi_{+},\Pi_{+}^{-1})$ and $\epsilon(\Pi_+)=1=\epsilon(\Pi_+^{-1})$.

=-1 One can notice that analogously to the previous case the algebra $U_{q,\tau}(\operatorname{iso}(g))$ can be seen as an algebra over polynomial ring $\mathbb{C}[\frac{1}{\kappa}]$. By assigning the numerical value to $\kappa$ in formulas – we are introducing new (non-isomorphic) Hopf algebra over $\mathbb{C}$, which constitutes the so-called specialized ($q$-analog) version $U_{q,\tau}(\operatorname{iso}(g))$ for the case of $\tau^2=0$. Again, one 