Ph.D Thesis | |

Ph.D Student | Sebbag Daniel |
---|---|

Subject | On the Structure and Properties of Supergroup-Theoretical Categories |

Department | Department of Mathematics |

Supervisor | PROF. Shlomo Gelaki |

An important family of fusion categories is the family of *group-theoretical*

*categories*, that is, categories which are defined by data from
group theory.

This family contains the categories Rep(*G*) (the category of finite
dimensional *G*-modules for a finite group *G*) and all the pointed
fusion categories (i.e., fusion categories whose simple objects are invertible).

Group-theoretical categories play an important role in the classification

of larger families of fusion categories. One such result is the
classification of nondegenerate braided fusion categories, as given in [DGNO1].
In their work, the authors prove that every nondegenerate braided fusion
category that contains Rep(*G*) as a maximal symmetric fusion subcategory
is the center of some group-theoretical category.

In this work, we extend the family of group-theoretical categories to the

class of finite tensor categories. We define *supergroup-theoretical
categories*,using data from the theory of supergroups. Using this family, we
extend some results from fusion categories to finite tensor categories. In
particular,we classify non-degenerate finite braided tensor category which
contains a Lagrangian subcategory braid equivalent to sRep(*W*), which is
the "opposite" case to non-degenerate braided fusion categories with
a Lagrangian subcategory braid equivalent to Rep(*G*), which are fully
classified.

Section 1 is devoted for general background in the theory of finite tensor

categories and superalgebras.

In Section 2, we study general properties of tensor categories associated

with supergroups.

In Section 3, we extend the family of pointed fusion categories to the

class of finite tensor categories. First, we define and study the
categories of representations of finite dimensional supercommutative quasi-Hopf
superalgebras (which we denote by sVec(*G**⋉**W, ω*) for *G**⋉**W *a finite
supergroup and *ω *a *G *3-cocycle). Then, we classify the
indecomposable exact module categories over these categories, and finally we
classify necessary and sufficient conditions for a finite pointed tensor
category to be equivalent to a category of the form sVec(*G**⋉**W, ω*).

In Section 4, we define supergroup-theoretical categories, study their

structure and properties, and show that some of the properties of group-

theoretical categories also hold for supergroup-theoretical categories. In

addition, we prove that all these categories have the Chevalley property

(i.e., their simple objects generate a full fusion subcategory).

In Section 5, we classify braided finite tensor categories with a
Lagrangian subcategory braided equivalent to sRep(*W*) (the category of finite
dimensional ∧*W*-supermodules
with even morphisms) using the classification of non-degenerate braided fusion
categories containing sVec as a Lagrangian subcategory. In particular, the
classification of *nondegenerate *braided finite tensor categories with a
Lagrangian subcategory braided equivalent to sRep(*W*) is a first step
toward extending the classification given in [DGNO1] to the nonsemisimple case.

In Section 6, we study the structure of the center of a supergroup-theoretical
category *C*. We prove that the center of a supergroup-theoretical category
is not supergroup-theoretical, but it does contains a supergroup-theoretical
category *D *as a full tensor subcategory, and *D *is uniquely
determined by *C*. We also describe the basic structure of *Z*(*C*)
(i.e., the simple objects and their projective covers).