Infinitude of Haagerup–Izumi categories over abelian groups

Determine whether there exist infinitely many finite abelian groups G for which the Haagerup–Izumi fusion ring associated to G admits a unitary categorification as a Haagerup–Izumi fusion category.

Background

The authors study actions of Haagerup–Izumi fusion categories on noncommutative 2-tori. A Haagerup–Izumi category is a unitary categorical realization of a specific fusion ring built from an abelian group G, with simple objects G ∪ {gρ}_{g∈G} and prescribed fusion rules.

While such categories are known in many cases, it remains unknown whether there are infinitely many abelian groups G admitting such unitary categorifications. The authors explicitly note this longstanding open problem in the context of their constructions.