All Questions
15 questions
12
votes
0
answers
552
views
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 ...
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 \...
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 ...
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:
...
34
votes
1
answer
3k
views
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 ...
6
votes
2
answers
256
views
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 ...
- 3,001
0
votes
1
answer
191
views
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 ...
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 $\...
3
votes
1
answer
145
views
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 ...
5
votes
1
answer
260
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
0
answers
251
views
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 ...
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 ...
41
votes
3
answers
3k
views
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 $...
- 43.2k
39
votes
9
answers
10k
views
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," "...
13
votes
4
answers
3k
views
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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