can easily re-scale the momenta $P^{a}$, $P^{-}$ by $\frac{1}{\kappa}$, so in fact, one can prove that different values of the parameter $\kappa$ give rise to isomorphic Hopf-algebras. Additionally, more general version of this construction is possible provided that one starts up from any basis in the associate vector space with $e_0=\tau$.

From the considerations presented in this section one can deduce that any universal enveloping algebra twisted by an (extended) Jordanian twist admits specialization of the deformation parameter to a numerical value.

$\boldsymbol{\kappa(\tau)}$-Minkowski spacetime
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Quantum $\kappa$-Minkowski spacetime $\mathcal{M}_{\kappa,\tau}^{D}$ is usually defined as an algebra (complex, universal, unital and associative) generated by the following relations [@koma95; @LLM] $$\begin{gathered}
\left[{x}^{\mu},{x}^{\nu}\right]=\frac{i}{\kappa}\left(\tau^{\mu}{x}^{\nu}-\tau^{\nu} {x}^{\mu}\right)
\label{kMtau}\end{gathered}$$ of Lie algebra type[^12], where $\tau^{\mu}$ are (real numerical) contravariant components of the vector $\tau \in V$ with respect to some basis $\{e_{\mu}\}_{\mu=0}^{D-1}$.

On one hand the algebra  is well adopted to the following action of the quantum $U_{\kappa,\tau}(\operatorname{iso}(g))$ $$\begin{gathered}
P_{\mu}\triangleright {x}^{\nu}=-\imath \delta_{\mu}^{\nu},
\qquad
M_{\mu \nu}\triangleright {x}^{\rho}=i\left(g_{\mu \alpha}\delta_{\nu}^{\rho}-g_{\nu \alpha}\delta_{\mu}^{\rho}\right){x}^{\alpha}.
\label{kM1}\end{gathered}$$ This implies $\Pi_{\tau}^{\pm 1}\triangleright {x}^{\mu}={x}^{\mu}\mp \frac{i}{\kappa}\tau^{\mu}$. On the linear subspace spanned by the generators $\operatorname{lin}_\mathbb{C}\{x^1,\ldots$, $x^D\}$ this action is equivalent to the (complexified) vector representation . The way it extends to the polynomial expressions $x^{\mu_1}\cdots x^{\mu_k}$ is controlled by 