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15
Finitely semisimple spherical categories and modular categories
, 2008
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GRAPHICAL METHODS FOR TANNAKA DUALITY OF WEAK BIALGEBRAS AND WEAK HOPF ALGEBRAS
"... Abstract. Tannaka duality describes the relationship between algebraic objects in ..."
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Abstract. Tannaka duality describes the relationship between algebraic objects in
Tannaka Reconstruction of Weak Hopf Algebras in Arbitrary Monoidal Categories
, 2009
"... We introduce a variant on the graphical calculus of Cockett and Seely[2] for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A − → B from a monoidal category A ..."
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We introduce a variant on the graphical calculus of Cockett and Seely[2] for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A − → B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannaka reconstruction gives a weak bialgebra in B; if, moreover, A is autonomous, this weak bialgebra is in fact a weak Hopf algebra. 1
MODULE CATEGORIES OF FINITE HOPF ALGEBROIDS, AND SELFDUALITY
"... Abstract. We characterize the module categories of suitably finite Hopf algebroids (more precisely, ×Rbialgebras in the sense of Takeuchi [Tak77] that are Hopf and finite in the sense of [Sch00]) as those klinear abelian monoidal categories that are module categories of some algebra, and admit d ..."
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Abstract. We characterize the module categories of suitably finite Hopf algebroids (more precisely, ×Rbialgebras in the sense of Takeuchi [Tak77] that are Hopf and finite in the sense of [Sch00]) as those klinear abelian monoidal categories that are module categories of some algebra, and admit dual objects for “sufficiently many ” of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be selfdual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them [Pfe09]. 1.
BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
a MaxPlanckInstitut für Mathematik
, 807
"... This is part one of a twopart work that relates two different approaches to twodimensional openclosed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the pro ..."
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This is part one of a twopart work that relates two different approaches to twodimensional openclosed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the properties of these algebras and prove uniqueness and existence theorems. One implication is that under certain natural assumptions, every rational closed CFT is extendable to an openclosed CFT. The relation of Cardy algebras to the solutions of the sewing constraints is the topic of part two. Contents 1 Introduction and summary 2 2 Preliminaries on tensor categories 6 2.1 Tensor categories and (co)lax tensor functors...................... 6 2.2 Algebras in tensor categories............................... 9
FIBER FUNCTORS, MONOIDAL SITES AND TANNAKA DUALITY FOR BIALGEBROIDS
, 2009
"... What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phùng Hô Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstr ..."
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What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phùng Hô Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed
Contents
, 2006
"... Abstract. In this article, we introduce and study Hopf monads on monoidal categories with duals. Hopf monads generalize Hopf algebras to a nonbraided (and nonlinear) setting. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence ..."
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Abstract. In this article, we introduce and study Hopf monads on monoidal categories with duals. Hopf monads generalize Hopf algebras to a nonbraided (and nonlinear) setting. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke’s criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again
BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
ON THE FIELD ALGEBRA CONSTRUCTION
, 806
"... Abstract. A pure algebraic variant of John Roberts ’ field algebra construction is presented and applied to bialgebroid Galois extensions and certain generalized fusion categories. 1. ..."
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Abstract. A pure algebraic variant of John Roberts ’ field algebra construction is presented and applied to bialgebroid Galois extensions and certain generalized fusion categories. 1.