# Tagged Questions

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### On the set of indices of irreducible depth 3 subfactors

Let $I_n$ be the set of indices of (finite index) irreducible depth $n$ subfactors. Then $I_2 = \mathbb{Z}_{>0}$. Question 1: Is it true that $I_3$ has no accumulation point? If so: ...
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### On a quantum Riemann Hypothesis

Robin's theorem (1984) states that $$\sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann hypothesis is true. Recall that $γ$ is the Euler–Mascheroni ...
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### Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
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### Can a planar tangle have an infinite number of input disks?

Can a planar tangle have an infinite number of input disks? Some publications talk about cases with a finite number of input disks, while others do not say if it is finite or infinite. So, is it ...
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### A Schur-like product theorem on groups

Let $G$ be a finite group, and consider the composition $X * Y$ on $\mathbb{C}G$ defined by $$(\sum_g \alpha_g u_g) * (\sum_g \beta_g u_g) = \sum_g \alpha_g \beta_g u_g.$$ This composition can be ...
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### Is the Euler characteristic of a subfactor planar algebra, nonzero?

Let $\mathcal{P}$ be an irreducible subfactor planar algebra and $\mu$ the Möbius function of its biprojection lattice $[e_1,id]$. Then the Euler characteristic of $\mathcal{P}$ is defined as follows: ...
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### Generalization of the product formula on subfactors

The product formula on finite groups states that for $H_1, H_2$ subgroups of $G$, then $$|H_1H_2| \cdot |H_1 \cap H_2|=|H_1| \cdot |H_2|$$ This statement could be generalized to any finite index ...
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### Can we build a subfactor planar algebra from one knot?

From one finite group $G$, we can build the subfactor $(R \subset R \rtimes G)$ which remembers the group. Question: Can we build a subfactor planar algebra from one knot? which remembers the knot? ...
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### Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here). Now ...
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### Is every irreducible subfactor planar algebra a quotient of the planar algebra of tangles?

Let $\mathcal{T}_{n,\pm}$ be the vector space generated by the planar tangles (up to isomorphism) having $2n$ intervals on their "ouput'' disk and a white (or black) shaded marked interval. Then the ...
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### Can any finite lattice with at most six elements be realized as an intermediate subfactor lattice?

The paper Lattices of Intermediate Subfactors of Y. Watatani, received on December 1994, finishes by: Prop. 6.2. $\$ Any finite lattice with at most five elements can be realized as an ...
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### What's the relation between fusion and coproduct?

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 ...
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### The coproduct on the 2-boxes space of the group-subgroup subfactor planar algebras

Let $(H \subset G)$ be an inclusion of finite groups. Let the subfactor $(\mathcal{R} \rtimes H \subset \mathcal{R} \rtimes G)$ with $\mathcal{R}$ the hyperfinite ${\rm II}_1$ factor, and its planar ...
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### Jordan-Hölder theorem for planar algebras?

First recall the Jordan-Hölder theorem for groups: Theorem (Jordan-Hölder): Let $G$ be a group, and let $$G=G_1 \supset G_2 \supset \dots \supset G_r = \{ e \}$$ be a normal tower such that ...
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### Planar algebraic translation of a subfactor property

Let $N \subset M$ be an irreducible finite depth and finite index subfactor. $M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows : M=\bigoplus_{...
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### Are every finitely generated planar algebras, also singly generated?

Let $\mathcal{P}$ be a finitely generated planar algebra. Question : Is it also singly generated ? I ask this question, because, on one hand I've read on this paper of V. Jones and D. Bisch : "...
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### An embedding theorem for a fusion ring planar algebra?

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 ...
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### Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Question: Are there only finitely many maximal irreducible amenable subfactors at ...
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### Invertibility of the planar algebra-subfactor correspondence

In Jones's paper "Planar Algebras I", Theorem 4.2.1 establishes that an extremal finite index subfactor admits a spherical C*-planar algebra structure, and Theorem 4.3.1 establishes that spherical C*-...
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### What are some natural and attractive open problems in Jones's theory of planar algebras?

I'm hoping to learn something about planar algebras while attacking a planar algebra question with an undergrad research student. I'm thinking about reading this paper, as Kuperberg's program seems ...
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### Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
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### Why are fusion categories interesting?

In the same vein as Kate and Scott's questions, why are fusion categories interesting? I know that given a "suitably nice" fusion category (which probably means adding adjectives such as "unitary," "...
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### ubiquity, importance of path algebras

I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...