[Paper Review] Weakly group-theoretical and solvable fusion categories

Pavel Etingof, Dmitri Nikshych+1 more|ArXiv.org|Sep 17, 2008

Algebraic structures and combinatorial models25 references16 citations

TL;DR

This paper introduces and studies weakly group-theoretical and solvable fusion categories over ℂ, establishing that all weakly group-theoretical categories satisfy the strong Frobenius property—meaning the ratio of the category's Frobenius-Perron dimension to that of any simple object in a module category is an algebraic integer. It further proves that fusion categories of dimension $p^r q^s$ are solvable, generalizing Kaplansky’s 6th conjecture and extending results on Hopf algebras and modular categories.

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.

Motivation & Objective

• To define and characterize weakly group-theoretical and solvable fusion categories as generalizations of group-theoretical and nilpotent categories.
• To establish a Morita equivalence criterion for fusion categories equivalent to group extensions of a given category via their Drinfeld centers.
• To prove that all weakly group-theoretical fusion categories satisfy the strong Frobenius property, extending Kaplansky’s 6th conjecture.
• To show that fusion categories of dimension $p^r q^s$ (with $p,q$ prime) are solvable, generalizing known results on small-dimensional Hopf algebras.
• To provide structural insights into fusion categories and semisimple Hopf algebras of prime power and composite dimensions, particularly $pqr$ and $pq^2$.

Proposed method

• Uses Morita equivalence and Drinfeld center theory to relate fusion categories to group extensions, employing the characterization that $\mathcal{C}$ is Morita equivalent to a $G$-extension of $\mathcal{D}$ iff $\mathcal{Z}(\mathcal{C})$ contains a Tannakian subcategory $\mathrm{Rep}(G)$ whose de-equivariantization yields $\mathcal{Z}(\mathcal{D})$.
• Applies de-equivariantization and equivariantization constructions to build sequences of fusion categories from $\mathrm{Vec}$ via cyclic groups of prime order.
• Employs the Müger centralizer and braided category techniques to analyze symmetric subcategories and detect nontrivial Tannakian subcategories in centers.
• Leverages the Frobenius-Perron dimension and algebraic integer properties to prove the strong Frobenius property in weakly group-theoretical categories.
• Uses induction and dimension counting arguments, particularly in the case of 2- and 4-dimensional simple objects in the center, to show the existence of Tannakian subcategories.
• Applies results from modular tensor categories and modularization (via Bruguières and Müger) to reduce problems to known classification theorems on small-dimensional categories.

Experimental results

Research questions

• RQ1 Does every fusion category satisfy the strong Frobenius property?
• RQ2 Are all weakly integral fusion categories weakly group-theoretical?
• RQ3 Is every fusion category of dimension $p^r q^s$ solvable?
• RQ4 Can a fusion category be Morita equivalent to a $G$-extension if its center contains a Tannakian subcategory $\mathrm{Rep}(G)$ with de-equivariantization equivalent to the center of the base category?
• RQ5 What structural properties do fusion categories of dimension $pqr$ or $pq^2$ possess, and are they group-theoretical?

Key findings

• A fusion category $\mathcal{C}$ is weakly group-theoretical if and only if it is Morita equivalent to a nilpotent fusion category, with a subsequent result showing such categories are Morita equivalent to group extensions of pointed categories.
• A fusion category $\mathcal{C}$ is Morita equivalent to a $G$-extension of $\mathcal{D}$ if and only if $\mathcal{Z}(\mathcal{C})$ contains a Tannakian subcategory $\mathrm{Rep}(G)$ such that de-equivariantizing the centralizer of $\mathrm{Rep}(G)$ yields $\mathcal{Z}(\mathcal{D})$ as a braided category.
• Any weakly group-theoretical fusion category has the strong Frobenius property: for any indecomposable $\mathcal{C}$-module category $\mathcal{M}$ and simple object $X \in \mathcal{M}$, the ratio $\mathrm{FPdim}(\mathcal{C}) / \mathrm{FPdim}(X)$ is an algebraic integer.
• All fusion categories of dimension $p^r q^s$ (with $p,q$ prime and $r,s \geq 0$) are solvable, generalizing known results for $pq$ and $p^r$ dimensions.
• Any non-pointed simple weakly group-theoretical fusion category is equivalent to $\mathrm{Rep}(G)$ for a finite non-abelian simple group $G$, and any fusion category of dimension 60 is weakly group-theoretical (hence, if simple, equivalent to $\mathrm{Rep}(A_5)$).
• All fusion categories of dimension $pqr$ (distinct primes) and all semisimple Hopf algebras of dimension $pqr$ or $pq^2$ are group-theoretical, and such Hopf algebras are classified via the framework developed in the paper.