All Questions
28 questions
22
votes
2
answers
2k
views
Non weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$ below)?
$$\scriptsize{\begin{...
17
votes
1
answer
2k
views
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-Popa ...
13
votes
1
answer
1k
views
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 ...
13
votes
3
answers
686
views
Does every Frobenius algebra in a monoidal *-category give a Q-system?
Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...
- 28.1k
9
votes
2
answers
356
views
Is there a subfactor construction involving 2-groups?
I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an ...
- 45.7k
9
votes
3
answers
409
views
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 ...
8
votes
0
answers
488
views
Is there a non-trivial Hopf algebra without left coideal subalgebra?
Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\...
8
votes
0
answers
306
views
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 ...
7
votes
2
answers
409
views
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 ...
- 2,441
7
votes
3
answers
698
views
Jordan-Hölder theorem for subfactors?
All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
7
votes
0
answers
169
views
How to translate connection on four graphs to quantum 6j symbols
I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...
- 437
6
votes
0
answers
128
views
Unitary fusion category and subfactor
From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor.
By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
6
votes
0
answers
239
views
Existence of a Kac algebra for a given fusion ring in a particular class
A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r}...
5
votes
1
answer
177
views
Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?
A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.
Is there an infinite depth irreducible finite index maximal subfactor (other than ...
5
votes
1
answer
203
views
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 unital inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth ...
5
votes
1
answer
261
views
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 \} $$ be a normal tower such that ...
4
votes
2
answers
542
views
The category of subfactors extending the category of groups?
This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
4
votes
0
answers
190
views
A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras
The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:
maximal,
it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
4
votes
0
answers
151
views
Is there a maximal finite depth infinite index irreducible subfactor?
A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...
3
votes
0
answers
138
views
Is there a finite depth irreducible subfactor of prime index and not group-subgroup?
Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
3
votes
0
answers
134
views
What are all the possible indices for the finite depth subfactors?
Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
3
votes
0
answers
111
views
Is there a non-irreducible maximal subfactor other than two-sided TLJ?
A subfactor $N \subseteq M$ is called:
irreducible if $N' \cap M = \mathbb{C}$,
maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$.
The two-sided ...
2
votes
0
answers
92
views
DHR superselection and DR reconstruction in low spacetime dimensions
Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator ...
- 437
2
votes
0
answers
89
views
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:
...
2
votes
0
answers
250
views
Fusion categories with permutation "associativity matrices"
Let $\mathcal{C}$ be a fusion category and let $(H_1,...,H_r)$ be its simple objects.
$\mathcal{C}$ is non-pointed if at least one of its simple object has Perron-Frobenius dimension $ \neq 1$.
...
2
votes
0
answers
166
views
How simplify the pentagonal equation from two fusion rings?
A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...
2
votes
0
answers
158
views
About the classification of infinite depth irreducible finite index maximal subfactors
The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group $SU(2)$....
1
vote
0
answers
174
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 $\...