Transmutation Theory and Quantization Approach for Quantum Groupoids

TL;DR

This paper extends transmutation theory and quantization methods for quantum groupoids, constructing new Hopf algebras in monoidal categories and exploring their relations, with implications for quantum algebra structures.

Contribution

It generalizes transmutation theory for quantum groupoids and constructs new Hopf algebras using quasitriangular structures and cocycles, advancing quantum algebra theory.

Findings

Construction of Hopf algebras in categories $_H\mathcal{M}$ and $_H\mathcal{M}_F$

Generalization of transmutation theory for quantum groupoids

Isomorphism between different Hopf algebra constructions

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 $C_{\widetilde H}(\widetilde{H}_s)$, 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 $(H, \mathcal{R})$.