Group-theoretical property of non-degenerate fusion categories of FP-dimensions p2q3 and p3q3

Abstract

In this paper, we show that non-degenerate fusion categories of FP-dimensions p 2 q 3 d and p 3 q 3 d are group-theoretical, where p, q are odd primes, d is a square-free integer such that (pq, d) = 1.

Group-Theoretical Property of Slightly Degenerate Fusion Categories of Certain Frobenius-Perron Dimensions

“…where (G,η) is a metric group of order p 2 . Thus the proof of [10,Theorem 3.3] shows that C(G,η) Z q can be embedded into a non-degenerate fusion category of Frobenius-Perron dimension p 2 q 2 , moreover, since we have assumed that C ad is not pointed, C(G,η) Z q contains a Tannakian subcategory of Frobenius-Perron dimension pq by Proposition 3.1 and [10, Theorem 3.3], so D is a group-theoretical fusion category Lemma 3.3. The subcase FPdim(D ad ) = 2pq 2 can be proved in the same way.…”

“…The following proposition is based on the classification results of [10], and it will play a key role in the classification theorems below. Proposition 3.1.…”

“…Proof. It follows from [10,Corollary 3.5] that C is a group-theoretical fusion category. Assume that C is not pointed below, let E be a maximal Tannakian subcategory of C, the existence of E is guaranteed by [6,Theorem 7.2].…”

“…Proof. If D ad ∩D ′ = Vec, then we deduce from Lemma 3.1 that D is split, so D is grouptheoretical by [10,Theorem 3.4]. Thus, we only need to consider the case when D ′ ⊆ D ad .…”

“…In particular, for odd primes p and q, slightly degenerate fusion categories of Frobenius-Perron dimension 2p m q n d are always integral and solvable [9, Corollary 3.4, Proposition 3.14], where d is an odd square-free integer such that (pq,d) = 1. Moreover, non-degenerate fusion categories C of Frobenius-Perron dimensions p 2 q 3 d and p 3 q 3 d are group-theoretical [10], that is, Z(C) are braided equivalent to Drinfeld centers of certain pointed fusion categories.…”