Transmutation Theory and Quantization Approach for Quantum Groupoids

Abstract

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 $C_{\tilde{H}}({\tilde{H}}_s)$, and obtain that they are isomorphic as objects in the category ${}_{\tilde{H}}\mathcal{M}$, where $(\tilde{H},\tilde{\mathcal{R}})$ is a new quasitriangular quantum groupoid induced by $(H,\mathcal{R})$.

20 pages

References in corpus (1)

• Yetter-Drinfeld modules over weak Hopf algebras and the center construction