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### Is there a subgroup of dual depth 3?

This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end... Let's ...
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### Is there a maximal subgroup of depth 3?

Let's first define what we mean by depth of a subgroup. Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
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### why can index be larger either the trace goes smaller or larger in index $\ge 4$ case?

I am studying the index of subfactor and basic construction recently. Suppose $M=\mathcal{R}$ to be the infinite hyperfinite II1 factor. For projection $p\in \mathcal{R}$, $p\mathcal{R}p$ is also a ...
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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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### Index of a subfactor of a full $II_1$ factor

On pg. 151 of "Coxeter Graphs and Towers of Algebras" by F.M. Goodman, P. de la Harpe, and V.F.R. Jones (1989), it is stated that there is no known example of a full $II_1$ factor having a subfactor ...
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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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### In what sense do Jones' original subfactors come from quantum SU(2)

In his paper Index for subfactors [Invent. Math., vol. 72 (1983), pp. 1-26], Vaughan Jones proved his remarkable index rigidity theorem, i.e., the fact that the possible index values for a (type II$_1$...
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### Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?

Let $S$ be the multiplicative semigroup of numbers generated by $B=\{ 2cos(\frac{\pi}{n}) \mid n \ge 4 \}$. Question: Does every number of $S$ factorize uniquely (up to perm.) as a product of ...
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### On the intersection of index 2 subfactors

Let $H_1$ and $H_2$ be two distinct index $2$ subgroups of a finite group $G$. We can deduce several properties about the intersection $H_1 \cap H_2$: $H_1$ and $H_2$ are normal subgroups of $G$. ...
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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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### Is the Nichols-Richmond theorem true for integral fusion rings?

The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper. It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here): Theorem: ...
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### Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)

To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post). A fusion ring is a finite dimensional complex space $\mathbb{C}\mathcal{B}$ together ...
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### Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?

A fusion ring is a finite dimensional complex space $\mathbb{C}\mathcal{B}$ together with a distinguished basis $\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $h_i \cdot h_j = \sum_k n_{ij}^kh_k$...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-... 1answer 120 views ### Is the coproduct of central operators, also central? Let$\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}^{-...
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 ...
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 ...