series in $\frac{1}{\kappa}$. The Lie sub-algebra corresponding to the stability group of $\tau_{+}$ consists of $\operatorname{iso}(p-1,q-1)=\operatorname{gen}\{M_{ab},M_{+b}\}$, i.e. the generators with the primitive coproducts. One can notice that $P_+$ and $\Pi_+$ are not algebraically independent and one can express one through another: $P_{+}=\kappa(\Pi_+-1)$ which will be helpful in introducing the $q$-analog version of the above Hopf algebra.

**$\boldsymbol{q}$-analog version of $\boldsymbol{U_{\kappa,\tau}(\operatorname{iso}(g))}$: $\boldsymbol{\tau^2=0}$.** Similarly to the previous section we start from considering the Hopf sub-algebra generated by the following elements $(M_{+-},M_{+a},M_{-a},M_{ab},P_{a}$, $P_{-},\Pi_{+},\Pi_{+}^{-1})$ and we denote it as before as $U_{q,\tau}(\operatorname{iso}(g))$. The only difference is that now $\tau^2=0$ and one deals with non-standard (triangular) deformation. It is generated by the following relations $$\begin{gathered}
 [M_{+ a},M_{- b} ]=-i (M_{ab}+g_{ab}M_{+ -} ),
\qquad
 [M_{\pm a},M_{\pm b} ]=0,
\label{Mpm_q}
\\
 [M_{\pm a},M_{bc} ]=i (g_{ab}M_{\pm c}-g_{ac}M_{\pm b} ),
\qquad
 [M_{+ -},M_{\pm a} ]=\pm iM_{\pm a},
\\
 [M_{+ -},\Pi_{+} ]=i (\Pi_{+}-1 ),
\qquad
 [M_{+ -},P_{-} ]=-iP_{-},
\qquad
 [M_{+ a},P_{b} ]=i\kappa g_{ab} (\Pi_{+}-1 ),
\\
 [M_{-a},P_{-} ]=0,
\qquad
 [M_{-a},P_{+} ]=-iP_{a},
\qquad
 [M_{-a},\Pi_{+} ]=-\frac{i}{\kappa}P_{a},
\\
 [M_{-a},P_{b} ]=ig_{ab}P_{-},
\qquad
 [M_{+a},\Pi_{+} ]= [M_{+a},P_{+} ]= [M_{+ -},P_{a} ]=0.\end{gathered}$$ The commutation relations within the $(D-2)$-dimensional sector $(M_{ab},P_{a},g_{ab})$ stay standard, cf. , . Another way would be, to abstractly define the algebraic structure as universal 