A construction of Hopf algebras in braided monoidal category by weak Hopf algebras

Details A construction of Hopf algebras in braided monoidal category by weak Hopf algebras A construction of Hopf algebras in braided monoidal category by weak Hopf algebras

2011-11-06

arxiv.org/abs/1111.1397v1

Mathematics

Rings and Algebras

23 pages

Scientific paper

Let $(H, \mathcal{R})$ be a quasitriangular weak Hopf algebra, $A$ a weak Hopf algebra, and $f$ a weak Hopf algebra map between $H$ and $A$. Then we show that $A$ induce a Hopf algebra $C_{A}(A_{s})$ in the category ${}_{H}\mathcal{M}$, which generalizes the transmutation theory introduced by Majid. Furthermore, we construct a Hopf algebra $C_{H}(H_{s})_F$ in the category ${}_H\mathcal{M}_F$ for any cocommutative weak Hopf algebra $H$ and a weak invertible unit 2-cocycle $F$, which generalizes the result in [5]. Finally, we consider the relation between $C_{H}(H_s)_{F}$ 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 weak Hopf algebra induced by $(H, \mathcal{R})$.

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