Pinhas Grossman

Quantum Subgroups of the Haagerup Fusion Categories

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 irreducible 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 irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS

A quadrilateral of factors is an irreducible inclusion of factors

Cyclic extensions of fusion categories via the Brauer–Picard groupoid

N \subset M

Strong singularity for subfactors

with intermediate subfactors

Intermediate subfactors with no extra structure

P

The Brauer-Picard group of the Asaeda-Haagerup fusion categories

and

The Asaeda–Haagerup fusion categories

Q

Quantum doubles of generalized Haagerup subfactors and their orbifolds

such that

Forked Temperley–Lieb algebras and intermediate subfactors

P

Fusion Categories Associated to Subfactors with Index

and