Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

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7
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1answer
1k 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 ...
11
votes
0answers
608 views

Non-“weakly group theoretical” integral fusion categories?

Can you exclude integral fusion categories of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices (I don't write the trivial one) ...
4
votes
3answers
386 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 ...
4
votes
0answers
102 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 ...
6
votes
2answers
278 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 ...
10
votes
3answers
907 views

A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form : $$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that : $d_{i}\vert n$ $1=d_{1}<d_{2} \le d_{3} \le ... \le ...
7
votes
3answers
261 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 ...
5
votes
1answer
180 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
0answers
164 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}, ..., ...
0
votes
1answer
113 views

Gradings Induced by Coactions?

A "well-known" fact is that a ${C}_q[U(1)]$-coaction on a vector space $V$ induces a $Z$-grading. I don't see why this is so. Clearly, a ${C}_q[U(1)]$-grading will induce a $Z$-grading. But why does ...
41
votes
11answers
4k views

What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
32
votes
6answers
3k views

What does “quantization is not a functor” really mean?

The answers to this question do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by ...
29
votes
4answers
2k views

Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
8
votes
2answers
1k views

Is there a machinery describing all the irreducible representations ?

Suppose we have a finite dimensional Lie algebra $g$, Is there a machinery to describe all the irreducible representation of $g$. Consider toy example: $sl_{2}$ or $sl_{3}$, how do we describe all ...
24
votes
1answer
1k views

Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
15
votes
2answers
1k views

How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
10
votes
1answer
1k views

Grothendieck and Non-commutative Geometry?

When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the ...
10
votes
3answers
1k views

How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?

I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
8
votes
0answers
234 views

Are there only finitely many maximal subfactors of a 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 subfactors of a fixed finite ...
4
votes
1answer
893 views

Homotopic morphisms between curved A-infinity algebras

I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...
6
votes
1answer
525 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
4
votes
0answers
160 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 ...
3
votes
1answer
271 views

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the bracket $$[xt^r, ...
3
votes
1answer
312 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 ...