# Existence of a Kac algebra for a given fusion ring in a particular class

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).

• I'm confused about why you're restricting to things satisfying those conditions. Just because any such Hopf algebra would fail to be maximal, doesn't mean that any Hopf algebra exists. – Noah Snyder Sep 10 '13 at 23:45
• @NoahSnyder : I'm not sure to well-understand your comment. In fact here, I'm just interested in (non-trivial) maximal irreducible depth 2 subfactors (i.e. non-trivial maximal Kac algebras), and the four conditions above are necessary for that, so I don't need to investigate those whose don't check these conditions. Now it's true that these conditions are not sufficient, and for example, the Kac algebra of a finite (non abelian) simple group checks them. If you know some additional necessary conditions, I'm very interested. – Sebastien Palcoux Sep 11 '13 at 12:37
• But none of your questions (comes from a Kac algebra, categorifiability) have anything to do with maximality. – Noah Snyder Sep 11 '13 at 13:51
• @NoahSnyder : You're right, but I progress step by step. First, a list of necessary conditions (for maximality), and a question about the existence of the Kac algebras (checking these conditions). The next step will be to find among these Kac algebras, those which are maximal. – Sebastien Palcoux Sep 11 '13 at 14:10
• @NoahSnyder : another point is that it's very difficult for me, to know if my example at dimension 210, is categorifiable or not (by solving the pentagonal equation, very hard...), so I thought that maybe all the fusion rings (including my example) checking these conditions, are automatically related to a Kac algebra. I checked it's true by a computer calculation, for dim<210, rank<9, and I ask here about a possible general proof. – Sebastien Palcoux Sep 11 '13 at 14:22