& = & 1 \otimes x_0 - a_0 ((1 - \lambda) \cdot 1 \otimes x_k p_k + \lambda x_k p_k \otimes 1),  \label{idx0}\end{aligned}$$ where $Z=e^{A}$. Let us name the relations (\[idxi\]),(\[idx0\]) obtained in this way by ${\mathcal{R}} $. Similarly as before, these relations induce the partition of the algebra ${\mathcal{A}} \otimes {\mathcal{A}} $ into equivalence classes, which together with the algebra $\Delta (T) $ forms the algebra $ \Delta(H) = \Delta({\mathcal{A}})\Delta(T)  $. This algebra contains all deformed coproducts $\Delta h, $ for all $h \in H. $ Again $\Delta h $ can be understood as the operator acting on $ \Delta(H) $ and acquiring values in the same algebra, namely $ \Delta h :  \Delta(H) 
 \longrightarrow  \Delta(H) . $ In this case the twist operator is defined accordingly as $ \mathcal{F} :  \Delta_0(H)
 \longrightarrow  \Delta(H) $, and vice versa for the inverse twist $\mathcal{F}^{-1}$.

By twisting the undeformed coproducts according to Eq.(\[defcops\]), one finds the following deformed coproducts $$\begin{aligned}
 \label{defcopxi}
  \Delta x_i & =& x_i \otimes Z^{\lambda} =Z^{\lambda-1}\otimes x_i, \\
  \Delta x_0 & =& x_0 \otimes 1 + a_0 (1 - \lambda)\cdot 1 \otimes x_k  p_k =1\otimes x_0 -a_0\lambda x_kp_k \otimes 1, \label{defcopx0} \\
   \Delta p_{i} &=& p_{i}\otimes Z^{-\lambda }+Z^{1- 