M.Sc Student | Joseph Shtok |
---|---|

Subject | On Group-Theoretical Categories |

Department | Department of Mathematics |

Supervisor | Full Professor Gelaki Shlomo |

Full Thesis text |

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 G−graded 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.