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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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### Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
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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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### Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
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### Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
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### Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M$ is irreducible if $N' \cap M = \mathbb{C}$. It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. It's cyclic if its lattice of ...
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### Are subfactor planar algebras hard to classify at index 6?

Given a finite index inclusion, $N\subset M$, of $II_1$ factors we can construct two towers of finite dimensional algebras known as the $\textit{standard invariant}$. For low index, this has allowed ...
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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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### 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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### 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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### An upper bound for the maximal subgroups at fixed index?

Let us call a subgroup an injective homomorphism between groups. I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone. A subgroup $H \subset G$ is ...
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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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Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}'$ are also a ${\rm II}_1$-factors. Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{... 0answers 191 views ###$\mathcal{L}(H_i \subset G_i)$distributive$\Rightarrow\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$modular? Let$\mathcal{L}( G)$be the lattice of subgroups of$G$and$\mathcal{L}(H \subset G)$the lattice of intermediate subgroups. Definitions: A lattice$(L, \wedge, \vee)$is distributive if,$\...
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Many people I talk to lament the nonexistence of a coherent source for learning the theory of subfactors. Could someone suggest a nice (ordered) list of books/papers to work through to obtain a ...
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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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### 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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### Isomorphism theorem for subfactors?

It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors : Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
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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....
Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-... 0answers 140 views ### The planar algebra generated by the biprojections Let$(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)$. Let$\...
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?