#### Date of Award

Spring 2007

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Dmitri Nikshych

#### Abstract

We give necessary and sufficient conditions for two pointed categories to be dual to each other with respect to a module category. Whenever the dual of a pointed 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, o), where G is a finite group and o ∈ H3(G, kx). A group-theoretical and cohomological interpretation of this relation is given. As an application, we give a series of concrete examples of pairs of groups that are categorically Morita equivalent but have non-isomorphic Grothendieck rings. 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.

The notion of a nilpotent fusion category, which categorically extends the notion of a nilpotent group, was introduced by Gelaki and Nikshych. We give sufficient conditions for a group-theoretical category to be nilpotent.

We classify Lagrangian subcategories of the representation category of a twisted quantum double Do( G), where G is a finite group and o is a 3-cocycle on it. This gives a description of all braided tensor equivalences between twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(Do( G)) and module categories over the category VecwG of twisted G-graded vector spaces such that the dual fusion category is pointed. As a consequence, we establish that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.

#### Recommended Citation

Naidu, Deepak, "Morita equivalence for group -theoretical categories" (2007). *Doctoral Dissertations*. 381.

https://scholars.unh.edu/dissertation/381