---
abstract: 'We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.'
address:
- 'Faculty of Engineering Sciences, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium'
- 'Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China'
- 'Department of Mathematics, Shandong Institute of Architecture and Engineering, Jinan, Shandong 250014, China'
author:
- 'S. Caenepeel'
- Dingguo Wang
- Yanmin Yin
title: 'Yetter-Drinfeld modules over weak bialgebras'
---

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Introduction {#introduction .unnumbered}
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Weak bialgebras and Hopf algebras are generalizations of ordinary bialgebras and Hopf algebras in the following sense: the defining axioms are the same, but the multiplicativity of the counit and comultiplicativity of 