H, and show that the category of Yetter-Drinfeld modules is isomorphic to

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Abstract. In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule ( = endodistributor) of a monoidal V-category A. When A is autonomous ( = rigid = compact), he showed that the V-category (that we call Tamb(A)) of soequipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A, V] of the convolution monoidal V-category [A, V]. Our paper extends these ideas somewhat. For general A, we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence [DA, V] ≃ Tamb(A). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V-category Tambs(A) (respectively, Tambls(A)) which is equivalent to the centre (respectively, lax centre) of [A, V]. We construct localizations DsA and DlsA of DA such that there are equivalences Tambs(A) ≃ [DsA, V] and Tambls(A) ≃ [DlsA, V]. When A is autonomous, every Tambara module is strong; this implies an equivalence Z[A, V] ≃ [DA, V]. 1.

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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,

Abstract. We prove that the Drinfeld double of the category of sheaves on

Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of [2] and [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule

Abstract. A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore–Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given an (eventually non-semisimple) k-linear abelian ribbon braided category C with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surf to k-vector spaces. 1991 Mathematics Subject Classification: 18B30, 18D10, 57N05 Moore and Seiberg’s study [16] on conformal field theory was continued and developed by Walker [23] from the topological point of view. The first systematic study in this direction of the example of ̂ sl(2) was made by Kohno [5, 6]. A different topological approach was proposed by Reshetikhin and Turaev [18] (see also [21]). The aim of this article is to present a categorical point of view on the subject. We use freely notations and results from the previous papers [9, 10]. We consider a category of surfaces S labeled by a set C, which Grothendieck calls the Teichmüller’s tower. Its subcategory Surf consists of labeled surfaces and isotopy classes of their homeomorphisms. Its central extension is denoted ESurf. We give also a definition of a modular functor. We show that any ribbon abelian category C, satisfying the axioms of modularity [10] yields a modular functor ZC: ESurf → k-vect. Thus, such category deserves to be called modular. Precisely, modularity means the following: C is a noetherian abelian k-linear ribbon tensor category with finite dimensional k-vector spaces HomC(A, B). In a cocompletion of C there exists a Hopf algebra F = ∫ X∈C

Abstract. A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore–Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given an (eventually nonsemisimple) k-linear abelian ribbon braided category C with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surf to k-vector spaces. 1991 Mathematics Subject Classification: 18B30, 18D10, 57N05 Moore and Seiberg’s study [16] on conformal field theory was continued and developed by Walker [23] from the topological point of view. The first systematic study in this direction of the example of ̂ sl(2) was made by Kohno [5, 6]. A different topological approach was proposed by Reshetikhin and Turaev [18] (see also [21]). The aim of this article is to present a categorical point of view on the subject. We use freely notations and results from the previous papers [9, 10]. We consider a category of surfaces S labeled by a set C, which Grothendieck calls the Teichmüller’s tower. Its subcategory Surf consists of labeled surfaces and isotopy classes of their homeomorphisms. Its central extension is denoted ESurf. We give also a definition of a modular functor.