[Paper Review] Classification of integral modular categories of Frobenius-Perron dimension pq^4 and p^2q^2

Paul Bruillard, César Galíndo+6 more|arXiv (Cornell University)|Mar 19, 2013

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper classifies integral modular categories of Frobenius-Perron dimension $pq^4$ and $p^2q^2$ for distinct primes $p$ and $q$. It shows that such categories are group-theoretical except when the dimension is $4q^2$, in which case non-group-theoretical examples arise from centers of Tambara-Yamagami categories or quantum groups, with a complete classification up to twist-equivalence for the $36$-dimensional case.

ABSTRACT

We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.

Motivation & Objective

• To classify all integral modular categories of Frobenius-Perron dimension $pq^4$ and $p^2q^2$ for distinct primes $p$ and $q$.
• To determine whether such categories are group-theoretical or not, especially in the case of dimension $4q^2$.
• To identify and classify non-group-theoretical integral modular categories of dimension $4q^2$, including those arising from quantum groups and Tambara-Yamagami categories.
• To provide a twist-equivalence classification for the $36$-dimensional non-group-theoretical modular categories arising from $ar{rak{sl}}_3$ quantum groups.

Proposed method

• Use of the classification of fusion subcategories of ${Rep}(D^\omega(G))$ for finite groups $G$ and 3-cocycles $\omega$ to analyze group-theoretical categories.
• Application of the bijection between modular structures on a nondegenerate braided fusion category and isomorphism classes of invertible self-dual objects to reduce the classification problem to braided categories.
• Leveraging the fact that Frobenius-Perron dimensions of simple objects divide the total dimension in integral modular categories to constrain possible fusion rules.
• Use of twist-equivalence and $G$-grading techniques to relate categories like $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ to known modular categories.
• Construction of new fusion categories $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ via twisting of $\mathcal{C}(\frak{sl}_3,q,6)$ using a normalized 2-cocycle $\chi$, preserving fusion rules.
• Proof that $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ is not group-theoretical by contradiction using central exact sequences and classification of semisimple Hopf algebras of dimension 12.

Experimental results

Research questions

• RQ1 Are all integral modular categories of dimension $pq^4$ group-theoretical?
• RQ2 What are the non-group-theoretical integral modular categories of dimension $4q^2$, and how can they be classified?
• RQ3 Is the $36$-dimensional modular category $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ twist-equivalent to a known quantum group category?
• RQ4 Can a $\mathbb{Z}_3$-graded fusion category with the same fusion rules as $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ be equivalent to $\mathrm{Rep}(H)$ for some Hopf algebra $H$?
• RQ5 What is the role of twist-equivalence in classifying modular categories beyond equivalence of braided fusion categories?

Key findings

• All integral modular categories of dimension $pq^4$ are group-theoretical.
• All integral modular categories of dimension $p^2q^2$ with $p^2q^2$ odd are group-theoretical.
• Non-group-theoretical integral modular categories of dimension $4q^2$ are equivalent to either $\mathcal{E}(\zeta,\pm)$ for an elliptic quadratic form $\zeta$ on $\mathbb{Z}_q \times \mathbb{Z}_q$, or are twist-equivalent to $\mathcal{C}(\frak{sl}_3,q,6)$ or $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$.
• The $36$-dimensional category $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ is a new example of a non-group-theoretical modular category not equivalent to $\mathrm{Rep}(H)$ for any Hopf algebra $H$, as shown via obstruction to abelian exact sequences.
• The modular data for $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ with $q = e^{\pi i/3}$ is explicitly computed, including the $S$ and $T$ matrices.
• The category $\bar{\mathcal{C}}(\frak{sl}_3,q,6)$ is not group-theoretical, as it cannot arise from a $\mathbb{Z}_3$-extension or equivariantization of a group-theoretical category due to the non-group-theoretical nature of $\mathcal{C}(\frak{sl}_3,q,6)$.