For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X\_i)\_{i \in I}$ of depth $0$ or $2$ correspond to the projections $p\_i = I\_{H\_i}$ on the $2$-boxes space $\mathcal{P}\_2(N \subset M) = \bigoplus\_{i \in I} End(H\_i) $ as an algebra.  

*Warning*: $I$ is the index set for the simple objects of depth $0$ or $2$, but there can have others. 


On a $2$-boxes space, there is a structure of coproduct $a∗b$ (defined for example [here][1] p4). 


**Question**: How compute the coproduct $p\_j \* p\_j$ from the fusion $X\_i \boxtimes X\_j$? 


My guess is that $p\_i \* p\_j \sim \sum\_{k \in K} p\_k$ with $K=\{ k \in I \ \vert \ X\_k \le X\_i \boxtimes X\_j \}$ (*truncated fusion*).  

Is it true? What's the exact formula? How prove it? 


Else, if there is a minimal projection $u \preceq p\_i∗p\_j$ such that its central support $Z(u)=p\_k$, but $p\_k \not \preceq p\_i∗p\_j$, is it still true that $X\_k$ appears as a summand of $X\_i ⊠ X\_j$?  

In other words, is it true that $Z(p\_i \* p\_j) \le \sum\_{k \in K} p\_k$? Is it an equality? (*weak truncated fusion*)



What's the relation between fusion and coproduct?
In the formula, $tr(b)$ should be replaced by $tr(|b|)$, i.e., $\|b\|\_1$. 


$$b\_\alpha\*b\_\beta=\sum\_\gamma\frac{\|b\_\alpha\|\_1\|b\_\beta\|\_1\bar{c}\_{\alpha\beta}^\gamma}{\sqrt{n}\|b\_\gamma\|\_1}b\_\gamma.$$


Unfortunately the proof needs the irreducibility. I do not know how to generalize it to weak Kac algebras. 


As Dave mentioned, one direction is clear. It is hard to determine when $(p\_i\*p\_j)p\_k=0$.


If $P\_{2,\pm}$ are abelian, then I have a criterion for the extremal case: $(p\_i\*p\_j)p\_k \neq 0$ for all permitted fusion rule if and only if $P$ has a Yang-Baxter relation (see [this paper](http://arxiv.org/abs/1507.06030)).


