Questions tagged [planar-algebras]
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57 questions
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Subfactors with integer Jones index
Is there any integer (Jones) index subfactor which is not extremal?
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111
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Inclusion of finite dimensional C*-algebras and relative commutants of subfactors
Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...
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Depth of the reduced subfactor
Suppose $N\subset M$ is a finite depth subfactor with $[M:N]<\infty$. Consider the reduced subfactor $pNp\subset pMp$ for some projection $p\in N$. How to calculate the depth of $pNp\subset pMp$ in ...
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Watatani's theorem for tensor categories
We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]:
Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has ...
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What about Hopf algebra and fusion structures for intertwiner algebras?
Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex
representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...
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On a revised quantum Riemann hypothesis
This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...
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132
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Complete list of indecomposable representations of Temperley-Lieb algebras at roots of unity?
The Temperley Lieb algebra $TL_n$ at roots of unity is not semisimple. The standard representations $V_{n,p}$ are indecomposable but, in general, not irreducible. If $K_{n,p}$ is the sub-...
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A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras
The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:
maximal,
it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
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Why is Planar algebras I (by Vaughan Jones) not published?
On Saturday 4 September 1999, Vaughan Jones put on arXiv a paper entitled Planar algebras, I.
Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the ...
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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
Here is a revised version: On a revised 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 ...
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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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On index 2 and square of subfactors without extra intermediate
Let $N \subsetneq K_i \subsetneq M$, $i=1,2$, be a square of irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, such that there is no extra intermediate, with $K_1 \not \...
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Existence of a third intermediate if there are two intermediate subfactors of index 2
Let $(N \subset M)$ be an irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors.
Let $K_1$ and $K_2$ be two distinct intermediate subfactors $N \subset K_i \subset M$, such ...
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191
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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 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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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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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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What's the detailed proof of "the composition of planar tangles is well-defined"?
In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...
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Are the two-side TLJ subfactors maximal?
Let $(N \subset M)$ be a unital 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 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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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 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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108
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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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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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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):
Question: ...
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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 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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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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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 ...
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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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What are the first non-maximal non-group-subgroup simple irreducible subfactors?
Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$
is normal if the biprojections $e_{\...
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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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Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?
The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
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What's the natural equivalence of subfactors in general?
Let $A$ be a factor and $\mathcal{C}_{A}$ be the category of all the subfactors $(M \subset N)$ such that $M$ and $N$ are isomorphic to $A$. The most famous of them is perhaps $\mathcal{C}_{R}$ with $...
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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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Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?
For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions:
• The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra $...
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What is the subfactor planar algebra of type $\tilde{A}_n$, of index 4?
As I understand it, there is a subfactor whose principal graph is the affine Dynkin diagram $\tilde{A}_n$. Since every vertex has two neighbors, does that mean the space of 1-boxes is two dimensional? ...
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