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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 ...
Sebastien Palcoux's user avatar
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 \...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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: ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Andi Bauer's user avatar
  • 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 ...
Sebastien Palcoux's user avatar
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 $\...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 $...
André Henriques's user avatar
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.
Elisha Peterson's user avatar