Integral almost square-free modular categories

Abstract

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension p n m, where p is a prime number, m is a square-free natural number and gcd(p, m) = 1. We prove that if n ≤ 5 or m is prime with m < p then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than 1125 is group-theoretical.

On the Classification of Almost Square-Free Modular Categories

“…Section 5]. In fact, it is shown in [9] that the core of C is a pointed non-degenerate braided fusion category.…”

“…Let C be an ASF modular category of Frobenius-Perron dimension dq n . In the case when C is integral, the structure of C has been obtained in [9,Corollary 4.2]: C is either pointed, or equivalent to a G-equivariantization of a nilpotent fusion category of nilpotency class 2, where G is a q-group. Note in addition that if C is strictly weakly integral, then q = 2, by Corollary 3.2.…”

Non-trivially graded self-dual fusion categories of rank 4

“…Combining the fact FPdim(C G ) = 1 |G| FPdim(C) with the fact Rep(G) = D 1 or D 2 , we get that FPdim(C G ) = 6 or 2. In both cases, C G is a pointed fusion category [7,Corollary 3.3]. It follows that C is a group-theoretical fusion category, by [16,Theorem 7.2].…”

Semisimple Hopf Algebras of Dimension 2q3

“…Let q be a prime number and let d be a square-free natural number. In [3] and [4] we call a fusion category almost square-free if its Frobenius-Perron dimension is dq n . In this paper we also call a Hopf algebra almost square-free if its dimension is dq n .…”