elements $h_1, h_2$ in $H,$ and has the properties $$\tau_0^{2}=1\otimes 1,  \qquad \tau_0\,\Delta_0 h=\Delta_0 h\,\tau_0 ,$$ for all elements $h$ in $H$. We use it to define the twist $\widetilde{\mathcal{F}}$ and the coproduct $\tilde{\Delta}$ $$\begin{aligned}
& \widetilde{\mathcal{F}}: \Delta_0(H) \longrightarrow
  \tilde{\Delta} (H),
& \qquad  \widetilde{\mathcal{F}}=\tau_0\, \mathcal{F}\,\tau_0,  \\
 &\tilde{\Delta}h : \tilde{\Delta} (H)
 \longrightarrow \tilde{\Delta} (H),
& \qquad \tilde{\Delta}h= \tau_0\, \Delta h\, \tau_0=
\widetilde{\mathcal{F}}\, \Delta_0h \, \widetilde{\mathcal{F}}^{-1},\end{aligned}$$ for every $h$ in $H$. Here $\widetilde{\mathcal{R}}=\tau_0 \mathcal{R} \tau_0 
 = \widetilde{\mathcal{F}} {\mathcal{R}}_0 \widetilde{\mathcal{F}}^{-1} $ are the relations coming from the coproduct $\tilde{\Delta}$ $$\begin{aligned}
&& x_i\otimes 1=Z^{\lambda}\otimes x_iZ^{1-\lambda}, \label{rtxi} \\
&& x_0\otimes 1=1\otimes x_0 +a_0\lambda\cdot 1\otimes x_kp_k +a_0(1-\lambda) x_kp_k\otimes 1  \label{rtx0}\end{aligned}$$ and $\tilde{\Delta} (H) = \tilde{\Delta} ({\mathcal{A}})
\tilde{\Delta} (T), $ where $\tilde{\Delta} ({\mathcal{A}})
={\mathcal{A}} \otimes {\mathcal{A}}/{\widetilde{\mathcal{R}}}.  $ The algebra $\tilde{\Delta} (H)$ contains all coproducts $\tilde{\Delta} h,$ for all $h$ in $H$. Note that which one among the relations ${\mathcal{R}}_0, {\mathcal{R}},
 \tilde{\mathcal{R}}$ needs to be applied in a given situation depends on the codomain of the operator in question.

The flip operator $\tau$ is defined with $\ \tau \, \Delta h=\Delta h \, \tau$. It follows that $$\label{tau}
\tau=\mathcal{F}\tau_0\mathcal{F}^{-1}= \tau_0\widetilde{\mathcal{F}}\mathcal{F}^{-1} =\tau_0R.$$ The last equality in is the defining relation of the $R$-matrix $$R=\widetilde{\mathcal{F}}\mathcal{F}^{-1},  \qquad  R: \ \Delta(H) \longrightarrow 