2005
DOI: 10.1007/bf02824824
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Yetter-Drinfeld modules over weak bialgebras

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 … Show more

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Cited by 41 publications
(13 citation statements)
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“…Remark 2.4. Note that when the ambient category C is symmetric and we take both the weak Yang-Baxter operator and the (M, D)-WO to be the braiding of C, we recover the classic definitions of Yetter-Drinfeld module introduced in [20] in the context of Hopf algebras and generalizated in [8] (see also [9] and [16]) to the context of weak Hopf algebras.…”
Section: The Category Of Yetter-drinfeld Modulesmentioning
confidence: 99%
See 1 more Smart Citation
“…Remark 2.4. Note that when the ambient category C is symmetric and we take both the weak Yang-Baxter operator and the (M, D)-WO to be the braiding of C, we recover the classic definitions of Yetter-Drinfeld module introduced in [20] in the context of Hopf algebras and generalizated in [8] (see also [9] and [16]) to the context of weak Hopf algebras.…”
Section: The Category Of Yetter-drinfeld Modulesmentioning
confidence: 99%
“…The theory of Yetter-Drinfeld modules for a weak Hopf algebra was introduced by Böhm in [8]. Later, Nenciu proved in [16] that this category is isomorphic to the category of modules over the Drinfeld quantum double (the interested reader can also see [9]).…”
Section: Introductionmentioning
confidence: 99%
“…The Drinfel'd double. The Drinfel'd double of a finite dimensional weak Hopf algebra H is studied in several papers [3,9,16]. Regarding weak bialgebras as bialgebroids (over separable Frobenius base algebras), the double construction of weak Hopf algebras fits Schauenburg's double construction of finite × R -Hopf algebras in [17].…”
Section: 1mentioning
confidence: 99%
“…Although it is possible to give the sufficient and necessary conditions, they are technically involved and so do not seem to be usable in practice. Our sufficient conditions, however, have a simple form and they are capable to describe the known examples (in particular the Drinfel'd double of a weak Hopf algebra [3,16,9]).…”
Section: Introductionmentioning
confidence: 99%
“…This is the motivation of the present article. In order to investigate these questions, we introduce the definition of Yetter-Drinfeld modules over weak Hom-Hopf algebras, which is generalization of both weak Yetter-Drinfeld modules introduced by [3] or [15] and Hom-Yetter-Drinfeld modules introduced by [10] or [11], and consider that when the Yetter-Drinfeld modules category of a weak Hom-Hopf algebra is is a rigid monoidal category, and is braided.…”
Section: Introductionmentioning
confidence: 99%