be removed by re-scaling: $\tau^\mu\mapsto
{\frac{1}{\kappa}}\tau^\mu$ in  (or alternatively $x^0\mapsto \kappa x^0$ in ). Such algebra $\mathcal{M}^D_{\tau}$ is isomorphic to the enveloping algebra of the solvable Lie algebra denoted usually[^14] as $\mathfrak{an}^{D}$. This version has found numerous applications, e.g. from the point of view of the spectral triples [@Andrea; @Sitarz3; @Sitarz2; @Sitarz1] or in the group field theories [@Oriti] which are connected with loop quantum gravity and spin foams approach.

Crossed product – unified description for DSR algebras
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The property  allows us to introduce a larger algebra which unifies $\operatorname{iso}(g)$ generators with that of $\mathcal{M}^D_{\kappa,\tau}$ by making use of a crossed product construction (see e.g. [@BPsigma]). This is the so-called DSR (deformed special relativity) algebra and it is based on the multiplication formula $$\begin{gathered}
(f\otimes L)\rtimes (g\otimes M)=f(L_{(1)}\triangleright g)\otimes L_{(2)}M
$$ providing the following crossed commutations $$\begin{gathered}
 [1\otimes L,f\otimes 1 ]_{\rtimes}=(L_{(1)}\triangleright f)\otimes L_{(2)}-f\otimes L\end{gathered}$$ between elements of two ingredient algebras: $\mathcal{M}_{\kappa,\tau}^{D}$ and $\mathcal{U}_{\kappa,\tau}(\operatorname{iso}(g))$ with the action induced by .

Therefore the covariant form of DSR algebra, generalized now to any metric $g$, any vector $\tau$ and living in arbitrary dimensions, obeys the algebraic relations , , supplemented by the following cross-commutation relations $$\begin{gathered}
[{P}_{\mu},{x}^{\rho}]_{\rtimes}=-\imath \delta_{\mu}^{\rho}\Pi_{\tau}+i\frac{\tau_{\mu}}{\kappa}{P}^\rho,
\\
[{M}_{\mu \nu},{x}^{\rho}]_{\rtimes}=i(g_{\mu \alpha}\delta_{\nu}^{\rho}-g_{\nu
