A cyclic subfactor is a subfactor admitting a distributive intermediate subfactors lattice.
Let's start with the finite index irreducible depth 2 subfactors, i.e. the class of subfactors of the form $(R^{\mathbb{A}} \subset R) $ with $R$ the hyperfinite II$_{1}$ factor and $\mathbb{A}$ a finite dimensional Kac algebra (Hopf C*-algebra).
By a theorem (here p 54) of Izumi-Longo-Popa, there is a (Galois) correspondence between the intermediate subfactors of $(R^{\mathbb{A}} \subset R) $ and the left coideal subalgebra of $\mathbb{A}$.
So $(R^{\mathbb{A}} \subset R) $ is a cyclic subfactor if and only if the left coideal subalgebras lattice of $\mathbb{A}$ is distributive. Let call such a Kac algebra, a cyclic Kac algebra.
Now, Izumi-Longo-Popa (p52) gives the following characterization of the left coideal subalgebras :
Definition. A unital von Neumann subalgebra $\mathbb{B}$ of a Kac algebra $\mathbb{A}$ is called a left (right) coideal von Neumann subalgebra if and only if $\Delta(\mathbb{B})\subset \mathbb{A}\otimes \mathbb{B}$ (respectively $\Delta(\mathbb{B})\subset \mathbb{B}\otimes \mathbb{A}$) holds.
Let $Corep(\mathbb{A})$ be the category of finite dimensional unitary corepresentations of $\mathbb{A}$.
Proposition. Let $\mathbb{A}$ be a compact Kac algebra. Then there exists one-to-one correspondence between the following two sets.
- The sets of left coideal von Neumann subalgebras of
$\mathbb{A}$.
- The set of systems of Hilbert subspaces $K_i \subset H_i$, $i \in Corep(\mathbb{A})$ satisfying the following:
$$K_i \oplus K_j \subset K_{i \oplus j}, \quad i, j \in Corep(\mathbb{A}).$$ $$K_i \otimes K_j \subset K_{i \otimes j}, \quad i, j \in Corep(\mathbb{A}).$$ $$\overline{K_i}=K_{\overline i}, \quad i \in Corep(\mathbb{A}).$$
Definition : We call the system $(K_{i})$ above, a ILP system (for Izumi-Longo-Popa).
A subfactor is maximal if it has no non-trivial intermediate subfactor. Then we call a Kac algebra "maximal" if it has no non-trivial left coideal subalgebra (obviously maximal $\Rightarrow$ cyclic).
Definition : Let $j \in Corep(\mathbb{A})$, then $v \in H_{j}$ is called fusion-cyclic if the ILP system it generates is the whole system, i.e., if $(K_{i})$ is a ILP system with $v \in K_{j}$, then $\forall i$, $K_{i} = H_{i}$.
Observation : $\mathbb{A}$ is a maximal Kac algebra if and only if $$\forall i \text{ non-trivial and } \forall v \in H_{i}, v \ne 0 \text{, then $v$ is fusion-cyclic.}$$
Question : Is it true that $\mathbb{A}$ is a cyclic Kac algebra if and only if $$\exists i \text{ (irr.?) and } \exists v \in H_{i} \text{ such that $v$ is fusion-cyclic ?}$$
Remark: Oystein Ore 1938 proved, on one page, that a finite group is cyclic iff its subgroups lattice is distributive. I hope its proof adaptable in the Kac algebra framework, for answering the question above.
Generalization to subfactors: Every finite depth, finite index, irreducible subfactor, is of the form $R^{\mathbb{I}} \subset R$, with $\mathbb{I} \subset \mathbb{A}$, a left coideal subalgebra of a finite dimensional weak Kac algebra $\mathbb{A}$ (see here).
The intermediate subfactors of $R^{\mathbb{I}} \subset R$ are given by the left coideal subalgebras $\mathbb{J} \subset \mathbb{I} \subset \mathbb{A}$.
Remark: I don't know if the proposition above is true in the weak case, I suppose it is.
Let $(L_i)$ be the ILP system of the left coideal subalgebra $\mathbb{I} \subset \mathbb{A}$.
Definition : Let $j \in Corep(\mathbb{A})$, then $v \in L_{j}$ is called $\mathbb{I}$-fusion-cyclic if the ILP system it generates is the system $(L_i)$, i.e., if $(K_{i})$ is a ILP system with $v \in K_{j}$, then $\forall i$, $K_{i} = L_{i}$.
Observation : $R^{\mathbb{I}} \subset R$ is a maximal subfactor if and only if $$\forall i \text{ non-trivial and } \forall v \in L_{i}, v \ne 0 \text{, then $v$ is $\mathbb{I}$-fusion-cyclic.}$$
Question : Is it true that $R^{\mathbb{I}} \subset R$ is a cyclic subfactor if and only if $$\exists i \text{ (irr.?) and } \exists v \in L_{i} \text{ such that $v$ is $\mathbb{I}$-fusion-cyclic ?}$$
Remark : A subfactor is also given by a fusion category and an algebra object. Then the lattice of intermediate subfactors is given by the lattice of subalgebras objects of the algebra object.
Is there a way to translate the question above in the language of the fusion categories ?