stability subgroups $G_\tau$ of the vector $\tau$[^4]. For a non-Euclidean metric it provides three non-isomorphic cases, which for the Lorentzian signature are the very well-known ones: vector $\tau$ can be time-, space- or light-like. In the Euclidean case there is only one $\kappa$-deformation[^5]. In the complex (signature independent) case one distinguishes two subcases instead: $\tau^2=0$ and $\tau^2\neq0$.

In the Drinfel’d quantization scheme one requires the so-called $h$-adic topology: extension of $U(\operatorname{iso}(g))$ by formal power series $U(\operatorname{iso}(g)) [[\frac{1}{\kappa}]]$ in order to arrange the deformation. It enables, e.g., existence of invertible twist, etc.(see, e.g., [@ChP; @Drinfeld1; @Drinfeld3] for more details). Hereafter for shortening the notation one introduces the following objects $$\begin{gathered}
\Pi_{\tau}=\frac{1}{\kappa}P_{\tau}+\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C},
\qquad
\Pi_{\tau}^{-1}=\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-\frac{1}{\kappa}P_{\tau}}{1+\frac{1}{\kappa^{2}}\left(\tau^{2}C-P_{\tau}^{2}\right)},
\\
\tau^{2}C_{\tau}=\kappa^{2}\left(\Pi_{\tau}+\Pi_{\tau}^{-1}-2+\frac{1}{\kappa^{2}}\left(\tau^{2}C-P_{\tau}^{2}\right) \Pi_{\tau}^{-1}\right)
=2\kappa^{2}\left(\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C} -1\right)\end{gathered}$$ as formal power series in ${\frac{1}{\kappa}}$, where $P_{\tau}=\tau^{\mu}P_{\mu}$. Moreover $C\equiv P^{\alpha}P_{\alpha}=g^{\alpha \beta}P_{\alpha}P_{\beta}$ denotes the well-known quadratic Casimir element (a.k.a.Casimir of mass in $D=4$ Lorentzian case). The element $C_{\tau}$ is also central and plays a role of deformed Casimir describing deformed dispersion relations (see, e.g., [@planck_scale1; @planck_scale2] and references therein). For the case $\tau^{2}=0$ one should take $C_{\tau}=C$. With this notation $\kappa(\tau)$-deformed (inhomogeneous) orthogonal Lie algebra, besides the standard orthogonal Lie algebra structure , , has deformed coalgebraic sector [@BP_unif] (cf. realization dependent form 