Deepak Naidu

Categorical Morita Equivalence for Group-Theoretical Categories

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 nonisomorphic 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.

Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double Dω(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(Dω(G)) and module categories over the category

Twisted quantum Drinfeld Hecke algebras

{\rm Vec}_{G} ^{\omega}

A Finiteness Property for Braided Fusion Categories

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 [D]. 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.

Fusion Subcategories of Representation Categories of Twisted Quantum Doubles of Finite Groups

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.

Some properties of group-theoretical categories

We generalize quantum Drinfeld Hecke algebras by incorporating a 2-cocycle on the associated finite group. We identify these algebras as specializations of deformations of twisted skew group algebras, giving an explicit connection to Hochschild cohomology. We classify these algebras for diagonal actions, as well as for the symmetric groups with their natural representations. Our results show that the parameter spaces for the symmetric groups in the twisted setting is smaller than in the untwisted setting.