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### On self-dual group-subgroup subfactors

Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction.
The subfactor $(N \subset M)$ is called self-dual if it is isomorphic to ...

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### On finite index infinite depth subfactors and reduction to depth 2

Let $(N \subset M)$ be a finite index irreducible subfactor (with $N$ and $M$, ${\rm II}_1$ factors).
Let $N \subset M \subset M_1 \subset M_2 \subset \cdots$ be the tower of basic constructions.
...

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### What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...

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### Is there no extra intermediate subfactor for the basic construction?

Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, the basic construction is $N \subset M \subset M_1 = \langle M , e^M_N \rangle$.
Question: For any intermediate subfactor $N \subset P \...

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### Does the Turaev-Viro theory for the generalized $E_6$ subfactor for $\mathbb{Z}/7$ distinguish $L(7,1)$ and $L(7,2)$?

In the paper Sato-Wakui "COMPUTATIONS OF TURAEV-VIRO-OCNEANU INVARIANTS OF 3-MANIFOLDS FROM SUBFACTORS" they compute certain Turaev-Viro-Ocneanu invariants of certain lens spaces. One of the results ...

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### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

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### Has a subfactor with lattice $B_3$, a singly generated identity biprojection?

Let $(N \subset M)$ be an irreducible finite index subfactor.
If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):
...

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### Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?

Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...

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### subfactor of finite rank but infinite index: is this possible?

A subfactor $N\subset M$ is essentially the same thing as an $N$-$M$-bimodule.
I'll recall the basic definitions in the language of bimodules, and I hope that subfactor people will excuse me.
...

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### Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
A subalgebra $I$ of $H$ is called a left coideal subalgebra if $\Delta(I) \subset H \otimes I$.
$H$ is ...

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### Short Introduction to Planar Algebras

Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.

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### Unitary structures on fusion categories

A unitary fusion category is a fusion category with a $C^*$-tensor structure.
Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more ...

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### What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ (...

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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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### Is 6 the smallest index for an irreducible subfactor to have a principal graph with a multiplicity >1 edge?

The irreducible subfactor $(R^{S_3} \subset R)$, of index $6$, admits a principal graph with a multiplicity $2$ edge because the group $S_3$ admits an irreducible complex representation of dimension $...

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### An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see here.
We are interesting in an alternative ...

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### Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra?
In other words, of the following form (...

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### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

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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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### Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states:
Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on $...

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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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### Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial.
proof: First if $...

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### Is there a non-solvable integral fusion category of square-free dimension?

A finite group of square-free order is solvable (see here).
You can find the definition for a solvable fusion category in this paper.
Question: Is there a non-solvable integral fusion category of ...

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### On the correspondence sub-N-N-bimodules and 2-box projections

Let $(N \subset M)$ be a finite index irreducible subfactor, and $P = P(N \subset M)$ its planar algebra.
We can see $M$ as an algebraic $N$-$N$-bimodule, it decomposes into irreducible algebraic $N$...

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### Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors.
Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ on $...

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### What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset &w\...

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### The completely reducible bimodules coming from subfactors

This post is a sequel of: Are all the R-R-bimodules completely reducible?
Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...

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### Is the (hyperfinite) TLJ subfactor unique at fixed index (if it exists)?

Let $(N \subset M)$ be an inclusion of hyperfinite ${\rm II}_1$ factors, with the following principal graph (called TLJ)
Question: Is such a subfactor unique (up to ${\rm W}^*$-isomorphism) at fixed ...

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### Are the two-side TLJ subfactors maximal?

Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, with the following principal graph (called two-side TLJ)
Question: Is $(N \subset M)$ a maximal subfactor?

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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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### Are there infinitely many amenable Hadamard-Petrescu subfactors?

The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by W....

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### A short problem with minimal projections and biprojections

Let $(N \subset M)$ be a finite index irreducible subfactor, $P=P(N \subset M)$ its planar algebra.
Notation: For $a,b \in P_{2,+}$ positive operators, then $\langle a,b \rangle$ is the biprojection ...

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### A process generating series of new subfactors

Consider the following process:
Take a maximal finite depth-index irreducible subfactor planar algebra $P^{(1)} = P(A^{(0)} \subset A^{(1)})$.
Choose a composition with itself such that there is no ...

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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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### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a single ...

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### Skein theory: How axiomatizing a 2-box space?

Let $(A,+,\times, *)$ with an adjoint operation compatible with $+$, $\times$ and $*$, such that $(A,+,\times)$ and $(A,+,*)$ are finite dimensional ${\rm C}^{*}$-algebras.
What are the axioms on $...

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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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### The planar algebra generated by the biprojections

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two irreducible finite index subfactors.
Let $\mathcal{B}_i$ be the set of all the biprojections of $\mathcal{P}_{2+}(N_i \subset M_i)$.
Let $\...

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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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### A section from subfactors to transitive groups

A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite II$...

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### Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...

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### Is there a tangle encoding the fusion rules?

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 ...

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### Is the coproduct of central operators, also central?

Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-...

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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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### Is there a Frobenius reciprocity for the coproduct?

Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-...

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### The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results:
- Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
- A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-...

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### Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor

Let $R$ be the hyperfinite type $III_1$ factor,
and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology
(topology of pointwise convergence on the predual).
An automorphism $\...

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### Are all the R-R-bimodules completely reducible?

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule.
Question: Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)?
Example: If $(N \...

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### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...

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### Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?

Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is ...