Printed: Oct 2016
We study unitary quotients of the free product unitary pivotal
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$-th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
In 1994, Bisch-Haagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
This is an abridged version of
pivotal category, free product, quotient, subfactor, intermediate subfactor
46L37 - Subfactors and their classification