Let $(N \subset M)$ be an irreducible finite index depth $n$ subfactor. Let $P = P(N \subset M)$ its planar algebra.
Let $(B_i)$ be the finite sequence of $N$-$N$-bimodules appearing in the principal graph.
Let $2m = n$ if $n$ even, else $2m=n+1$.
Let $p_i \in P_{2m,+}$ be the minimal central projection related to the $N$-$N$-bimodule $B_i$.

Question: Is there a planar tangle $T: P_{2m,+} \otimes P_{2m,+} \to P_{2m,+}$ such that $T(p_i \otimes p_j) = \sum_{k} n_{ij}^k p_k $ with $B_i \boxtimes B_j = \bigoplus_k M_{ij}^k \otimes B_k$ and $dim(M_{ij}^k)= n_{ij}^k$ (the fusion coefficients)?

Else, is there such a $T$ if we only consider the range support? the central support?

Remark: If $n = 2$, such a $T$ exists, it's the coproduct (see here).
Then, a generalization of the coproduct on $P_{2m,+}$ could do the job.

  • 1
    $\begingroup$ Is there a reason that you want to have all projections on the same number of strings. For example take $A_n$ than the objects are the Jones-Wenzl projections on $0,...,n-1$ strings. In this case I can give you the tangle. See page 97: books.google.com/… $\endgroup$ – Marcel Bischoff Dec 27 '14 at 18:16
  • $\begingroup$ @MarcelBischoff: I'm agree with this approach without the same number of strings, if it's more natural. $\endgroup$ – Sebastien Palcoux Dec 29 '14 at 6:58
  • $\begingroup$ I mean you can easily get a projection on $m+2n$ strings (with $n\in\mathbb N$) from a projection on $m$ strings. $\endgroup$ – Marcel Bischoff Dec 30 '14 at 20:19

I would think about something like this, where the caps and cups stand for $b$ strings the left one for vertical lines for $a$ strings and the right for $c$ strings. It has to be normalized to give again a sum of projections though. And this gives a map $P_{a+c}\otimes P_{c+b} \to P_{a+b}$. And it will not work if there are multiplicities $>1$.

enter image description here

Added Remark:

Note also that for finite depth, there is some $k$ such that $N\subset M_k$ is depth 2, where $M_k$ comes from the iterating Jones' basic construction. Then you can consider the planar algebra of $N\subset M_k$ (I guess the buzzword is cabling) and reduce to the depth 2 case. Then you can use the co-product. But, as far as I understand (see Zhengwei's answer here), this might cause problems, because $N\subset M_k$ is in general not irreducible.

  • $\begingroup$ Every fusion category is the tensor category of representations of a weak Hopf algebra (ENO). For the fusion category of the $N$-$N$ bimodules in the principal graph of $(N \subset M)$, it should be the weak Kac algebra coming from the depth $2$ subfactor $(N \subset M_k)$. $\endgroup$ – Sebastien Palcoux Dec 30 '14 at 21:27
  • $\begingroup$ Is the second sentence a question? $\endgroup$ – Marcel Bischoff Dec 30 '14 at 21:46
  • $\begingroup$ In the second sentence I write "it should be", because I did not check, but it would be natural to be this. Do you know if it's effectively this? $\endgroup$ – Sebastien Palcoux Dec 31 '14 at 8:35
  • 1
    $\begingroup$ Yes, if you choose $k$ right and choose the right weak Kac algebra from the two (dual to each other) associated to the subfactor. Note that the weak Hopf algebra is far from unique. A somehow canonical example you get if you take the object $A=\bigoplus_{i,j }M_i\boxtimes M_j$ ($\{M_i\}$ iso-classes of irreducible N-N bimodules in your fusion category), which has naturally the structure of a Q-system and gives you an reducible, self-dual subfactor $N\subset M_A$, which has depth 2, so a weak Kac algebra. $\endgroup$ – Marcel Bischoff Dec 31 '14 at 17:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.