Weakly group-theoretical and solvable fusion categories (0809.3031v2)

Abstract: We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.

• The paper introduces weakly group-theoretical and solvable fusion categories by connecting them to nilpotent and cyclically nilpotent structures via Morita equivalence.
• It demonstrates the strong Frobenius property, ensuring that the Frobenius-Perron dimension of any simple object divides the whole category's dimension.
• Furthermore, the authors prove that fusion categories with dimensions of the form p^r q^s are solvable, advancing classification methods in fusion category theory.

Overview of "Weakly Group-Theoretical and Solvable Fusion Categories"

The paper "Weakly Group-Theoretical and Solvable Fusion Categories" by Etingof, Nikshych, and Ostrik introduces and analyzes two classes of fusion categories: weakly group-theoretical categories and solvable categories. This research contributes to understanding the structure and properties of fusion categories, a central topic in the field of category theory.

Key Definitions

A fusion category is a rigid, semisimple tensor category with finitely many isomorphism classes of simple objects, finite-dimensional spaces of morphisms, and a simple unit object. The authors define a fusion category as weakly group-theoretical if it is Morita equivalent to a nilpotent fusion category. They introduce a solvable fusion category through two equivalent conditions: it is Morita equivalent to a cyclically nilpotent fusion category, or it is generated by a sequence involving cyclic groups of prime order through equivariantization and extension operations.

Main Results

The paper provides several critical results regarding weakly group-theoretical and solvable fusion categories:

1. Characterization of Morita Equivalence: Theorem 1.3 establishes conditions under which a fusion category is Morita equivalent to a G-extension, showing a deep connection between the structure of the Drinfeld center of fusion categories and their Morita equivalence classes.
2. Strong Frobenius Property: Theorem 1.5 asserts that any weakly group-theoretical fusion category satisfies the strong Frobenius property. This means that the Frobenius-Perron dimension of any simple object divides the entire category's Frobenius-Perron dimension.
3. Solvability of Fusion Categories: Theorem 1.6 shows that any fusion category with Frobenius-Perron dimension of the form $p^rq^s$ (where $p$ and $q$ are primes, and $r$ and $s$ are non-negative integers) is solvable. This result extends the scope of solvability in fusion categories to those with dimensions consisting of products of powers of two distinct primes.

Implications and Future Directions

These findings have several implications:

• Classification: The characterizations provided help in classifying fusion categories and understanding their underlying structure concerning group actions and extensions. The results about weakly group-theoretical categories pave the way for describing such categories in group-theoretical terms.
• Hopf Algebras: The conclusions extend to the paper of Hopf algebras, particularly those of dimensions matching the studied fusion categories. Notably, the authors further propose the classification of semisimple Hopf algebras of specific dimensions, which serve as concrete applications of their theoretical advancements.
• Modular Categories: The exploration of the Drinfeld center and related structures in fusion categories also impacts the theory of modular categories, with potential applications to topological quantum computing and conformal field theories.

The authors conclude with open questions, inviting further research into finding fusion categories not exhibiting the strong Frobenius property and exploring weakly integral but not weakly group-theoretical fusion categories. This paper lays a foundation for ongoing research in category theory and related areas in mathematical physics, with a particular focus on understanding and classifying fusion categories within the broader context of quantum groups and representation theory.