M.Sc Thesis
M.Sc Student Shtok Joseph On Group-Theoretical Categories Department of Mathematics Professor Shlomo Gelaki

Abstract

This work deals with a special family of a class of fusion categories,called the group-theoretical categories.Roughly speaking,fusion categories are those whose structure contains tensor product of objects,duals of objects,finite number of simple objects  and the property that every object decomposes as a direct sum of simple ones.Group-theoretical categories are defined by data , where H<G are finite groups, . Let be the category of Ggraded complex vector spaces with associator defined by the 3-cocycle . Then a group-theoretical category  is the subcategory of  generated by the bimodules of an associative algebra in the category. Main reason for the study of these categories lie in questions related to representation categories of semisimple Hopf algebras.

We study the structure of group-theoretical categories and compute Frobenius-Schur indicators for their simple objects. The indicators were first introduced in the context of finite groups and lately employed in various types of categories. They serve as fine categorical invariants, hence the computation of indicators in this context contributes to the study of semisimple Hopf algebras via their categories of representations.

Our main results are as follows:

1. We provide an explicit construction of a general simple object in the given category in terms of related subgroup of H and its projective representation.

2. Construction of dual objects and the related issues are treated.

3. The group Inv(C)  of invertible objects in group-theoretical category is computed. It is found to be isomorphic to the semidirect product

where the action of   on is given by conjugation.

4. A detailed treatment of the tensor powers of simple objects is carried out. The results are used in the computation of the Frobenuis-Schur indicators in the category.

5. Frobenius-Schur indicators are computed for a group-theoretical categories with a certain normalization of the 3-cocycle !. Explicit formulae are obtained for two

cases: (1) Second Frobenius-Schur indicator for any simple object in C. (2) General indicator for a simple object with cetrain assumption on its structure.

6. Examples include the indicators of the representation category of the twisted quantum double of a finite group G.