Ph.D Thesis
Ph.D Student Daniel Sebbag On the Structure and Properties of Supergroup-Theoretical Categories Department of Mathematics Full Professor Gelaki Shlomo

Abstract

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(GW, ω) for GW 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(GW, 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).