A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r})$ of their dimensions, its type, and the ring they generate for $\oplus$ and $\otimes$, its (integral) fusion ring.
Remark : $d_{1} = 1$ and $n=\sum d_{i}^{2}$.
Frobenius type hypothesis : We suppose that $d_{i} \vert n$ (it's a conjecture that it's always true).
Four conditions : There are some necessary conditions on the type $(1,d_{2},d_{3}, ..., d_{r})$ for the Kac algebra to be maximal (i.e., without non-trivial coideal subalgebra) and not $\mathbb{C}\mathbb{Z}_{n}$ (see this paper) :
- $n$ not of the form $p^{a}q^{b}$ or $pqr$, with $p, q, r$ prime numbers.
- $d_{2} \ge 3$ (in particular, a unique $1$-dimensional representation).
- at least three different dimensions (in particular rank $r \ge 3$)
- $gcd(d_{2},d_{3}, ..., d_{r}) = 1$ (it's a consequence of the Frobenius type hypothesis).
(If you know another necessary condition, please post it in comment)
Remark : We ask here about the finiteness of the number of such type for a fixed rank.
The concept of fusion ring can be defined for itself independently (see here p28).
A fusion ring is integral if the Perron-Frobenius of these basic elements, are integers.
A fusion ring is categorifiable if it is related to a fusion category. By Ocneanu's rigidity, a fusion ring is related to at most finitely many fusion categories, up to equivalence (see here).
Remark : There are non-integral fusion rings which are not categorifiable (even at rank $2$), and there are also non-categorifiable integral fusion rings (see corollary 7.4 here).
Experiment : I verified that every integral fusion ring of dimension $<210$, rank $<9$ and checking the four conditions above, is related to a Kac algebra (and in particular, is categorifiable).
Question : Is every integral fusion ring, checking these four conditions, related to a Kac algebra (or at least, is categorifiable) ?
Remark : If you know a counter-example, do you see also an additional "necessary condition" for excluding it ? Now, if it's true, it seems hard to prove, nevertheless, these four conditions are very restrictive, so maybe there is an "attainable" general proof.
The first undetermined example I know, is of dimension $210$ and rank $7$ (see this post).
