Transmutation Theory and Quantization Approach for Quantum Groupoids

Let $H$ and $L$ be quantum groupoids. If $H$ has a quasitriangular structure, then we show that $L$ induces a Hopf algebra $C_{L}(L_s)$ in the category $_{H}\mathcal{M}$, which generalizes the transmutation theory introduced by Majid. Furthermore, if $H$ is commutative, we can construct a Hopf algebra $C_H(H_s)_F$ in the category $_H\mathcal{M}_F$ for a weak invertible unit 2-cocycle $F$, which generalizes the results in \cite{D83}. Finally, we consider the relation between two Hopf algebras: $C_H(H_s)_F$ and $C_{\widetilde H}(\widetilde{H}_s)$, and obtain that they are isomorphic as objects in the category $_{\widetilde H}\mathcal{M}$, where $(\widetilde H, \widetilde{\mathcal{R}})$ is a new quasitriangular quantum groupoid induced by $(H, \mathcal{R})$.