Article

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

10/2004;

Source: arXiv

- [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non strict monoidal category. It is also shown that this category is not trivial, that is to say that it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.03/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**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 $(H, \mathcal{R})$.11/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**Given a weak distributive law between algebras underlying two weak bialgebras, we present sufficient conditions under which the corresponding weak wreath product algebra becomes a weak bialgebra with respect to the tensor product coalgebra structure. When the weak bialgebras are weak Hopf algebras, then the same conditions are shown to imply that the weak wreath product becomes a weak Hopf algebra, too. Our sufficient conditions are capable to describe most known examples, (in particular the Drinfel'd double of a weak Hopf algebra).05/2012;

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.