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Yetter-Drinfeld modules over weak bialgebras. (English) Zbl 1132.16031

Summary: 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.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16D90 Module categories in associative algebras
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