Article

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

10/2004;
Source: arXiv

ABSTRACT

We introduce Yetter-Drinfeld modules over a weak Hopf algebra \$H\$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of \$H\$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules 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.

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• "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]). "
##### Article: On the double crossed product of weak Hopf algebras
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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).
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• "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]). "
##### Article: The monoidal category of Yetter-Drinfeld modules over a weak braided Hopf algebra
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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.
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• "a well know fact that, if the antipode of a weak Hopf algebra H is invertible, H H YD is a non-strict braided monoidal category. In the following lines we give a brief resume of the braided monoidal structure that we can construct in the category H H YD (see Proposition 2.7 of [18] for modules over a field K or Theorem 2.6 of [12] for modules over a commutative ring). For two left-left Yetter-Drinfeld modules (M, ϕ M , M ), (N, ϕ N , N ) the tensor product is defined as object as the image of ∇ M ⊗N (see 1.5). "
##### Article: Quantum groupoids with projection
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ABSTRACT: In this survey we explain in detail how Radford's ideas and results about Hopf algebras with projection can be generalized to quantum groupoids in a strict symmetric monoidal category with split idempotents.