Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

We first recall the embedding theorem for finite depth subfactor planar algebras:
The planar algebra generated by a (finite depth) subfactor, is embeddable into the planar algebra generated by its principal graph.

In the same spirit, we could imagine to build a planar algebra generated by a fusion ring.
Then, the planar algebra generated by a (finite depth) subfactor could be embeddable into the planar algebra generated by the fusion ring of its principal even part.

Question : How generate a planar algebra from a fusion ring ?


Edit : we show how build a planar algebra from a fusion ring. Previously, we explain where this construction comes from, for showing it could be relevant. We finally ask about an embedding theorem :

In the finite index, depth 2 and irreducible case, a subfactor is of the form $R^{\mathbb{A}} \subset R$ with $\mathbb{A}$ a finite dimensional Kac algebra.

As C$^{*}$-algebra, a finite dimensional Kac algebra $\mathbb{A}$ admits finitely many irreducible representations
$H_{1}$, ..., $H_{r}$, of increasing dimension $n_{1} = 1$, $n_{2}$, ... , $n_{r}$.

The comultiplication $\Delta$ gives an action of $\mathbb{A}$ on $H_{i} \otimes H_{j}$ which decomposes as follows : $$ H_{i}\otimes H_{j} = \bigoplus_{k}{M_{ij}^{k} \otimes H_{k}} $$ with $M_{ij}^{k}$ the multiplicity space of dimension $n_{ij}^{k}$, so that : $\sum n_{i}.n_{j} = \sum n_{ij}^{k} . n_{k}$

The fusion ring coming from $\mathbb{A}$ is generated by the basis $(h_{i})$ and the relations : $$ h_{i}. h_{j} = \sum_{k}{n_{ij}^{k} h_{k}} $$

The inclusion matrix of $[\Delta(\mathbb{A}) \subset \mathbb{A} \otimes \mathbb{A}]$ is $\Lambda = (\Lambda_{(i,j)}^{k})$ with $\Lambda_{(i,j)}^{k} = n_{ij}^{k}$ (see this link).

Now, the inclusion matrix $\Lambda$ gives a bipartite graph $\mathcal{G}(\Lambda)$, and so a graph planar algebra $\mathcal{P}(\Lambda)$.

Question : Does $\mathcal{P}(R^{\mathbb{A}} \subset R)$ embed into $\mathcal{P}(\Lambda)$ ?

Generalization : Let $\mathcal{F}$ be a fusion ring with structure constants $(n_{ij}^{k})$, let $\mathcal{G}(\Lambda)$ be the bipartite graph of matrix $\Lambda = (\Lambda_{(i,j)}^{k})$ with $\Lambda_{(i,j)}^{k} = n_{ij}^{k}$. The fusion ring planar algebra generated by $\mathcal{F}$, is the graph planar algebra $\mathcal{P}(\Lambda)$ of $\mathcal{G}(\Lambda)$.

Question : If $\mathcal{F}$ comes from a subfactor $(N\subset M)$, does $\mathcal{P}(N\subset M)$ embed into $\mathcal{P}(\Lambda)$ ?

Remark : 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).

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.