A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. 


Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley-Lieb $A\_{\infty} $) ?



Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?
The infinite depth subfactor coming from SU(3) at any index above 9 gives an example. Here the Q-system is $V\_{(1,0)} \otimes V\_{(0,1)} \cong V\_{(1,1)} \oplus V\_{(0,0)}$ so the only possible sub-objects are the whole thing or the trivial, so it's certainly maximal.


