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**1 - 9**of**9**### From subfactors to categories and topology III. Triangulation invariants of 3-manifolds and Morita equivalence of tensor categories

- In preparation

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### A quantum double construction in Rel

, 2010

"... We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of ..."

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We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of

### YETTER-DRINFELD MODULES FOR TURAEV CROSSED STRUCTURES

, 2002

"... Abstract. We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of ..."

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Abstract. We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a Yetter-Drinfeld module over a crossed group coalgebra H and we prove that both the

### 3. The outer dual and the inner dual of a T-coalgebra 6

, 2002

"... Abstract. We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed ..."

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Abstract. We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed

### Contents

, 1994

"... 2. Braided categories and C-categories 3 2.1. C-categories 2.2. Braided categories ..."

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2. Braided categories and C-categories 3 2.1. C-categories 2.2. Braided categories

### ON MATRIX QUANTUM GROUPS OF TYPE An PHUNG HO HAI

, 1998

"... Abstract. Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the correspondin ..."

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Abstract. Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is “Zariski ” dense in the quantum group. Finally we give a formula for the integral. 1.

### unknown title

, 2010

"... We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we ..."

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We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we

### Dpto.

, 809

"... A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category C, and under certain ass ..."

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A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category C, and under certain assumptions on the braiding (fulfilled if C is symmetric), we construct a sequence for the Brauer group BM(C;B) of B-module algebras, generalizing Beattie’s one. It allows one to prove that BM(C;B) ∼ = Br(C) × Gal(C;B), where Br(C) is the Brauer group of C and Gal(C;B) the group of B-Galois objects. We also show that BM(C;B) contains a subgroup isomorphic to Br(C) × H 2 (C;B,I), where H 2 (C;B,I) is the second Sweedler cohomology group of B with values in the unit object I of C. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure R is contained in H and B is a Hopf algebra in the category HM of left H-modules. The Hopf algebras

### Tensor categories: A selective guided tour ∗

, 2008

"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."

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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on k-linear categories with finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way. 1 Tensor categories 1.1 Strict tensor categories