of the parameter matters and may lead to non-isomorphic Hopf algebras (e.g., for roots of unity) or influence representation theory. In our case a real value for this parameter is dictated by real forms of (complex) Hopf algebras under consideration which can be extended to module algebras as well. These issues are studied with more care.

In the following we shall introduce the ’$q$-analog’ version of $\kappa (\tau)$-deformation of non-semisimple inhomogenous orthogonal algebras $\operatorname{iso}(g)$, for all values of the vector $\tau$ with a fixed deformation parameter, i.e. $\kappa\in\mathbb{C}$. It appears that in this case one can re-scale all the formulas to get rid off the deformation parameter altogether. Firstly we recall the case of the orthogonal $D=1+(D-1)$ decomposition for time-like $\tau$ and introduce its $q$-analog version (analogous arguments will hold for space-like case as well). Later we consider the null-plane deformation and $D=2+(D-2)$ decomposition. One should underline that the $q$-analog version has been common for the standard deformations (i.e. for which $r$-matrix satisfies modified Yang–Baxter equation (MYBE), as in the time-like case). On the contrary it is rather unexpected for non-standard deformations (i.e. for which $r$-matrix satisfies classical YBE and when the twist exists, like in the light-like case). As 