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Séminaire Math-Physique : Morita equivalence of group-theoretical categories and modular invariants

Mercredi 22 novembre 2017 16:15-17:15 - Michaël Mignard - Université de Bourgogne

Séminaire Math-Physique : Morita equivalence of group-theoretical categories and modular invariants

Résumé : For a finite group G, we consider the category of finite dimensional G-graded vector spaces on a algebraic closed field of characteristic 0. Also, we twist the usual tensor product of vector spaces by a 3-cocycle w on G : we thus obtain a fusion category Vec(G,w) that is called pointed, that is any of its simple objects admits an inverse with respect to the tensor product. Inside such a category, one can extend the definion of algebras and modules as for the category of vector spaces (in fact, this is possible for any fusion category). For an algebra A in a general fusion category C, the category of bimomules over A in C is said to be Morita equivalent to C : for the case of Vec(G,w), such a category is described by a subgroup H of G and a 2-cochain on H which statisfies a compatibility relation with w. In this talk, we will discuss the classification of pointed fusion category up to Morita equivalence. A key tool is that two fusion categories are Morita equivalent if and only if their centers are equivlaent braided fusion categories. As it happens, the center of Vec(G,w) is furthermore a modular category equivalent to the category of representation of the Dijkgraaf-Pasquier-Roche twisted quantum double D(G,w). We use explicit
computations of some modular invariants of D(G,w)-Mod to obtain our classification.

Lieu : Salle A318

Pour en savoir plus sur cet événement, consultez l'article Séminaires Math-Physique