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The center construction for weak Hopf algebras. (English) Zbl 1029.16023

This paper discusses the center construction for weak Hopf algebras.
Let \(H\) be a weak Hopf algebra over a field \(k\). Denote by \(\text{Rep}(H)\) the category of representations of \(H\), whose objects are finite-dimensional left \(H\)-modules and whose morphisms are \(H\)-linear maps. Then the category \(\text{Rep}(H)\) is a monoidal category with unit object \(H_L\), and \(\text{Rep}(H)\) is braided if and only if \(H\) is a quasitriangular weak Hopf algebra. This paper defines the left-right Yetter-Drinfeld modules over a weak Hopf algebra \(H\) and the category \(_H{\mathcal{YD}}^H\) of the left-right Yetter-Drinfeld modules over \(H\). It is proved that \(_H{\mathcal{YD}}^H\) is a braided category, and that the category \(_H{\mathcal{YD}}^H\) can be identified with the category \(\text{Rep}(D(H))\) of left modules over the Drinfeld double \(D(H)\). Furthermore, this paper shows that the center \(Z(\text{Rep}(H))\) of \(\text{Rep}(H)\) for a weak Hopf algebra \(H\) is braided equivalent to \(\text{Rep}(D(H))\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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