operators in the representation space and are mutually inverse of each other. From one hand, these might be complicated issues involving numerical value of the parameter $\kappa$, domain of $P_{\tau}$, etc. From the other hand representations with a constant value for the mass Casimir operator $C$, e.g. irreducible ones, are very welcome.

From the point of view of applications in physical theories, specialization of the deformation parameter allows to interpret it as some physical constant of Nature, e.g quantum gravity scale $M_{\rm QG}$. However the value of it depends on a system of units one is using. For example, one should be able to use natural (Planck) system of units, $\hbar=c=1$. This fits very well with the re-scaling property mentioned above.

The $\boldsymbol{D=2+(D-2)}$ orthogonal decomposition: $\boldsymbol{\tau^2=0}$
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=-1 In the previous subsection we were dealing with the $q$-analog version of the standard $\kappa$-deformation (i.e. time-like case which includes the $\kappa$-Poincaré algebra). For $\tau^2\neq0$ the corresponding satisfies MYBE. It should be stressed that such $q$-analog version has not been considered for non-standard deformations (i.e. for which $r$-matrix satisfies CYBE and the cocycle twist exists[^10]). In this section we will show that, in fact, it is possible to construct the $q$-analog for 