Let Ꮿ be a fusion category faithfully graded by a finite group G and let Ᏸ be the trivial component of this grading. The center ᐆ(Ꮿ) of Ꮿ is shown to be canonically equivalent to a G-equivariantization of the relative center ᐆ Ᏸ (Ꮿ). We use this result to obtain a criterion for Ꮿ to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about the existence of zeroes in S-matrices of weakly integral modular categories.

Abstract. A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ∈ H 3 (G, k × ). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have non-isomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.

We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.

We describe all fusion subcategories of the representation category Rep(D ω (G)) of a twisted quantum double D ω (G), where G is a finite group and ω is a 3-cocycle on G.

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Abstract. We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3-cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(D ω (G)) and module categories over the category Vec ω G of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [Dr]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.

Abstract. Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.

Abstract. We classify integral modular categories of dimension pq 4 and p 2 q 2 , where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q 2 . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q 2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.