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2
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0answers
35 views

Noncommutative fusion categories

Although noncommutativeness is almost a defining trait for fusion categories, offhand I recall only the extended Haagerup N-N (rank 8). It's a two minute computation to find that even a based ring ...
-1
votes
0answers
91 views

Fusion: Hexagon implies pentagon?

Or in decent language: :-) Does the Racah backcoupling rule(s) imply Biedenharn-Elliot? Or for based rings: Can a based ring be braided and still not be a fusion category? (The 2-1-4-8 here would be a ...
1
vote
0answers
93 views

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 ...
3
votes
1answer
148 views

Fusion categories: If infinity were an integer

Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and ...
0
votes
0answers
36 views

Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-) I call a complex number $z$ pseudo-cyclotomic if $|z|=1$. I ...
1
vote
0answers
57 views

Automorphisms of Grothendieck rings coming from fusion categories

The Grothendieck ring $\mathcal K_0(\mathcal C)$ of a fusion category $\mathcal C$ is a unital based ring in the sense of math/0111139v1 with basis $B$ and involution $*$. Basis elements correspond to ...
5
votes
1answer
93 views

For what $G$ is $Rep(D(S_3))_{ad}$ Grothendieck equivalent to $Rep(G)$?

Given a fusion category $\mathcal C$, the Grothendieck Ring $K_0(\mathcal C)$ is the $\mathbb Z$-based ring whose basis elements correspond to isomorphism classes of simple objects and whose ...
0
votes
0answers
52 views

What is a fusion system/category without duals or an identity

What do you call a fusion system or fusion category without duals or an identity? For fusion systems I am using Definition 3.1 from On Arithmetic Modular Categories. I want to remove axioms (i) and ...
5
votes
1answer
251 views

Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
5
votes
1answer
123 views

What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$. ...
1
vote
0answers
86 views

When is the category of endofunctors braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...
4
votes
1answer
126 views

Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory ...
1
vote
0answers
69 views

Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$. Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...
3
votes
0answers
54 views

Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element: The square of $a$ equals Muger's "squared dimension" of $X$, an ...
2
votes
0answers
45 views

“Prime” fusion rings

Surely this concept is known! (But I don't recall seeing it - maybe under another name? But "prime" is the obvious name choice.) Example. Open the Gepner/Kapustin paper at ...
2
votes
0answers
208 views

Is there a non-trivial maximal Hopf algebra?

Let $H$ be a Hopf algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Maximal means without left coideal subalgebra $I$ (i.e. $\Delta(I) \subset H \otimes I$) other than ...
2
votes
1answer
107 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 ...
1
vote
1answer
143 views

“Generators” for fusion rings

It's a rather obvious idea in the area of fusion rings, but I haven't found a reference yet. Start with the usual rules for a rank n fusion ring $X_i\bigotimes{X_j}=\Sigma_k{T_{ij}}^kX_k$ and ...
2
votes
1answer
448 views

Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$. Questions: are there definitions of image and kernel for a ...
10
votes
1answer
270 views

A linear category with objects of infinite length but which is otherwise finite?

Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...
1
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0answers
67 views

Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
2
votes
1answer
90 views

No basis change in a fusion ring allowed?

Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$. A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4). ...
1
vote
0answers
128 views

The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual. Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...
7
votes
1answer
111 views

Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes?

Let $C$ be a fusion category with simple objects $X_1,...,X_n $, and let $Y_1,...,Y_n$ be objects with each $Y_i$ isomorphic to $X_i$. Is there a monoidal auto-equivalence $F:C \rightarrow C $ which ...
3
votes
1answer
70 views

Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...
1
vote
3answers
345 views

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 ...
2
votes
0answers
139 views

Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of Are there workable algebraic geometry approaches for the pentagon equation? I've replaced "algebraic geometry" by "numerical" in its content, ...
6
votes
1answer
629 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 ...
6
votes
2answers
226 views

How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be? The reason I am interested in this is that my ...
8
votes
1answer
179 views

Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?

Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...
5
votes
1answer
225 views

What is the exact meaning of “fusion” in the terminology “fusion category”?

I want to translate the terminology “fusion category” into Chinese, so I should know the exact meaning of "fusion". There are two translations in Oxford Advanced Learner’s Dictionary: 1.the process ...
2
votes
0answers
70 views

Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?)

Consider the category where objects are strict spherical fusion categories and morphisms are strict spherical functors (preserving cups and caps). I am wondering whether there is some kind of image ...
1
vote
0answers
198 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
0answers
140 views

Is an integral simple fusion ring, categorifiable?

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum ...
3
votes
1answer
111 views

Is every premodular category the *full* subcategory of a modular category?

In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore ...
7
votes
1answer
168 views

realizing fusion categories as subfactors of the hyperfinite

Let R be the hyperfinite II_1 or the hyperfinite III_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules. This question is inspired by the recent ...
4
votes
0answers
180 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 ...
4
votes
0answers
177 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}, ..., ...
1
vote
0answers
113 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 $\mathcal{R}_{1}$ ...
3
votes
0answers
73 views

Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...
7
votes
3answers
295 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 ...
2
votes
0answers
120 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 ...
12
votes
1answer
805 views

Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices? $\small{\begin{smallmatrix} 1 & ...
9
votes
0answers
302 views

Unitary structures on fusion categories

A unitary fusion category is a fusion category with a $C^*$-tensor structure. Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more ...
6
votes
1answer
364 views

Fusion category and Hopf algebra

Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed ...
5
votes
2answers
140 views

Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...
7
votes
1answer
277 views

If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?

If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if ...
12
votes
1answer
343 views

Are there interesting semisimple algebras in non-semisimple categories?

Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple ...
7
votes
1answer
190 views

Is there a fusion category with an object which does not commute with its dual?

Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)? Feel free to add adjectives such as pivotal, spherical, ...
12
votes
3answers
510 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 ...