I am in an automorphism quest.
Let $H$ be a quasitrianglar Hopf Algebra with R-matrix $\mathcal{R} \in H \otimes H$. I know that $\mathcal{R}_{21}^{-1 }$ is a solution of the Yang Baxter equation and $$\mathcal{R}_{21}^{-1} \Delta \mathcal{R}_{21} = \Delta^{op}.$$ Let $\cdotp$ be multiplication law on $H^{\star}$ defined as : \begin{equation} \label{eq1} \alpha \cdotp \beta = \langle \alpha_{(1)} \otimes \beta_{(1)}, \mathcal{R} \rangle \alpha_{(2)} \star \beta_{(2)} \end{equation} With $\star$ the convolution product and $\Delta^{\star} (\alpha) = \alpha_{(1)} \otimes \beta_{(2)}$ the comultiplication on $H^{\star}$ dual to the multiplication on $H$. The following exchange relation is true : $$ \beta \cdotp \alpha = \langle \alpha_{(1)} \otimes \beta_{(1)} R\rangle \langle \alpha_{(3)} \otimes \beta_{(3)} \mathcal{R}_{21}^{-1} \rangle \alpha_{(2)} \star \beta_{(2)} $$
The preceding exchange relation is invariant by the substitution $\mathcal{R}_{12} \rightarrow \mathcal{R}_{21}^{-1}$. This strongly indicates that the algebras obtained by choosing $\mathcal{R}_{12}$ or $\mathcal{R}_{21}^{-1}$ in the definition of $\cdotp$ are isomorphic.
How can I find such an isomorphism ? I would like to have an explicit formula.
