Quantum Subgroups of the Haagerup Fusion Categories

Abstract

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the "quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

The Asaeda–Haagerup fusion categories

“…(One may think of these embeddings as giving four independent constructions of the Extended Haagerup subfactor planar algebra.) Theorem 1.2 also connects the results of [Pet10] and [GS12] to complete the classi cation of graph planar algebra embeddings for the Haagerup planar algebra. In the last section of [Pet10], three embeddings of the Haagerup planar algebra into graph planar algebras were found, corresponding to the two principal graphs and the 'broom' graph.…”

“…However, it was not proven there could not be others. The main result of [GS12] shows there are exactly three module categories over the Haagerup subfactor planar algebra. Thus we have: Section 2 summarizes the combinatorial structure of the four Extended Haagerup fusion categories and the Morita equivalences between them.…”

Cyclic extensions of fusion categories via the Brauer–Picard groupoid

“…Both of these facts follow from Han's thesis [Han11]. Self-duality follows from uniqueness of the 2221 subfactor up to complex conjugacy, and no outer automorphisms follows from the explicit quadratic relations satisfied by Han's generators (as in [GS12b,Lemma 5.3] and [GS12a, Thm. 4.9]).…”

“…In fact, the Brauer-Picard 1-groupoid of the Izumi-Xu fusion category I 3 is a single point with automorphism group Z/2Z. Following the approach in [GS12b], the only possible minimal algebra objects in the Izumi-Xu fusion category are 1 and 1 + g + g 2 , and those each have unique algebra structures. Thus, the only simple module categories are I 3 and M 3 .…”

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

“…Finally we need to determine if there are any further unitarizable quotients of Fib ⊠N or T T ⊠N 3 . This is equivalent to classifying central commutative algebras in these categories, a hard problem in general [16]. However we are able to find arguments specific to the two above categories that bypass having to classify such algebras.…”