# Questions tagged [fusion-categories]

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159
questions

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### Modularity of the Drinfeld center of the category of G-graded vector spaces

Background: Let $G$ be a finite group, and $\mathrm{Vect}_G$ be the category of finite dimensional $G$-graded vector spaces over some algebraically closed field $k$ of char 0. It is well-known that $\...

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### Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules

Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules).
All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...

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### Are there interesting examples of unitary fusion categories where a tensor product of two simple objects is simple?

Let $\mathcal{C}$ be a unitary fusion category. Is it true that the tensor product of any two simple objects is simple ? If not, are there interesting (nontrivial) examples of such a $\mathcal{C}$ ?

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### Number of categories of product of fusion rings

Given two fusion rings $A$, $B$, and let $n_A$, $n_B$ be the number of gauge-inequivalent fusion categories belonging to $A$ and $B$. When one looks at the number of categories $n_{A\otimes B}$ of the ...

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### How does the Tannaka duality work for weak Hopf algebras and fusion categories?

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...

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### Does the pentagon axiom force the associativity constraint to be a natural isomorphism?

Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...

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### Questions about proof that all indecomposable module categories over $\operatorname{Rep}(G)$ are equivalent to $\operatorname{Rep}^1(H,\omega)$

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\End{End}$In Ostrik - Module categories, weak Hopf algebras and modular invariants, it is ...

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### Simple modular tensor category and zero entries in its S-matrix

Question 1: Is there a simple modular fusion category with a zero entry in its S-matrix?
(or equivalently, with a fusion matrix of zero determinant?)
Yes, by this answer below providing the example $\...

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### Relationship between fusion category and its Drinfel'd center

Is it true that given a fusion category $\mathcal{C}$ and its Drinfel'd center $Z(\mathcal{C})$, there is a fully faithful functor $F:\mathcal{C}\hookrightarrow Z(\mathcal{C})$? I.e. can $\mathcal{C}$ ...

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### Watatani's theorem for tensor categories

We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]:
Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has ...

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### Is the Frobenius property invariant by Morita equivalence?

Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....

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### Is there a non-pointed simple integral modular fusion category?

The weakly group-theoretical conjecture (supporting a negative answer to [ENO11, Question 2]) states as follows:
Statement 1: Every integral fusion category is weakly group-theoretical.
We wonder ...

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### 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 (...

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### Are the fusion categories weakly Frobenius?

A well-known open problem (generalizing Kaplansky 6th conjecture) asks whether every (spherical) fusion category $\mathcal{C}$ (over $\mathbb{C}$) is of Frobenius type, i.e. for every simple object $X$...

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### Vertex operator algebras and modular tensor categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C}=Rep(\mathcal{V})$ be the tensor category of (ususal) $\mathcal{V}$-modules. It is a well-known open-problem whether every ...

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### Different modular data with same T-matrix

Modular data (MD) is an invariant of a modular fusion category $\mathcal{C}$. It is a couple of symmetric matrices $S, T \in M_r(\mathbb{C})$ with:
$r$ the rank of $\mathcal{C}$,
$S$ invertible,
$T$ ...

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### What about Hopf algebra and fusion structures for intertwiner algebras?

Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex
representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...

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### Is there a strongly noncommutative Grothendieck ring?

This sequel of Is there a strongly noncommutative fusion category? is motivated to know whether every fusion category is "equivalent" (in some sense) to one with a commutative Grothendieck ...

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### Is there a strongly noncommutative fusion category?

A fusion category is called noncommutative if its Grothendieck ring is noncommutative. Let us call a fusion category strongly noncommutative if every fusion category Morita equivalent to it (i.e. same ...

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### Strongly simple fusion categories: the known examples?

A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same ...

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### Relation between the modular categories SU(2)_n and Sp(n)_1

The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...

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### Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]

Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...

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### Non-cyclotomic modular fusion categories

In a recent talk (see [1]) Richard Ng asks whether the invariants of 3-manifolds derived from any modular fusion category are cyclotomic integers. In tqft, such an invariant is computed from the F-...

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### Non-semisimple categorification problem of fusion rings

We refer to [1] for the notions used in this post.
The Grothendieck ring of a fusion category (over $\mathbb{C}$) is a fusion ring, but there are fusion rings which are not of this form (...

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### Hierarchy of fusion categories

There is a large hierarchy of foobar fusion categories, where foobar is a special property. How does this correlate with the properties of their Clebsch-Gordan (or analogue, I think those are the F-) ...

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### A twisted Haagerup category without pivotal structure

Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a quadratic ...

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### Sum of squares and divisibility

Consider an integer of the form $$N = 1 + \sum_{i=1}^r d_i^2$$ where $d_i \in \mathbb{N}_{\ge 3}$ and $d_i^2$ divides $N$.
Question: Must $r$ be greater than or equal to $9$?
Checking (with SageMath): ...

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### Categorical dimension and formal codegrees

Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of ...

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### Drinfeld center of a Deligne tensor product

Let $\mathcal{C}$ and $\mathcal{D}$ be two tensor categories (if necessary, assume they are fusion categories). Is the canonical braided monoidal functor $$\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(...

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### Is there a fusion subcategory in sphericalization tensor equivalent to the original one?

Let $C$ be a fusion category. Then $C$ is not necessary spherical. But its sphericalization $\tilde{C}$ has a canonical spherical structure $i:Id\to **$. The simple objects of $\tilde{C}$ are pairs $(...

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### The fusion categories with a strict skeleton

We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post.
A fusion category is skeletal if two isomorphic objects are always equal. Every ...

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### Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?

The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...

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### Pseudo-real (Frobenius-Schur indicator $= -1$) simple object in $X \otimes X^*$?

If I consider a simple object $X$ in a fusion category and tensor it with its dual $X^*$, and let $Y$ be a simple object in the decomposition $X \otimes X^* = I + Y + \dotsb$. I want to say that $Y$ ...

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### Realizing a fusion category as endomorphisms of an algebra

Consider the two statements:
"Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra", as stated in 1506.03546 page 4. The above ...

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### Second Frobenius-Schur indicator and near-group categories G+|G|

A near-group category $G+m$ is a (spherical) fusion category whose simple objects are given by the element $g$ of the finite group $G$, plus one extra simple object $y$, with Grothendieck ring as ...

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### 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 ...

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### Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?

Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...

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### Reference for NIM-rep theory for non-commutative fusion rings?

The literature on nonnegative integer matrix representations (NIM-reps) seems to be focused on commutative fusion rings, since a primary motivation there is for rational conformal field theory (RCFT). ...

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### Is the regular representation of a fusion ring a direct sum of all its irreducible representations?

For groups, the regular representation contains each irrep with multiplicity equal to the irrep's dimension.
For (not necessarily commutative) fusion rings, is there any analogous statement for the ...

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### A property forcing the Frobenius-Schur indicator to be positive

Let $G$ be a finite group. Two irreducible complex representations $V,V'$ of $G$ are called dual to each other if $V \otimes V'$ admits a trivial component, i.e. $\hom_G(V \otimes V',V_0)$ is positive ...

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### Are the finite quantum permutation groups, weakly group-theoretical?

Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...

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### Schur orthogonality relation on fusion categories

Let $\mathcal{F}$ be the Grothendieck ring of an abelian fusion category. Let $(M_i)$ be its fusion matrices and $(\mathrm{diag}(\lambda_{i,j}))$ their simultaneous diagonalization. Take $M_1=id$, so ...

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### Finite groups G with Rep(G) Grothendieck equivalent to a modular category

We refer to Chapter 8 of the book Tensor Categories for notions related to modular tensor categories and J.P. Serre for the basic theory of linear representations of finite groups over $\mathbb C$.
...

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### Local fusion categories

A local fusion category ${\cal R}$ is a unitary fusion category equipped with a top-faithful surjective monoidal functor to the fusion category of vector spaces: $\beta: {\cal R} \to {\cal V}ec$. Here,...

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### Existence of twisted metaplectic categories

The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $SO(N)_2$, $N&...

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### Interpolated simple integral fusion categories of Lie type

$\DeclareMathOperator\PSL{PSL} \DeclareMathOperator\Rep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $\Rep(G(q))$, with $G(q)$ ...

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### Extension of a theorem of Bisch to cyclotomic integers of fixed degree

Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...

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### R-matrices and symmetric fusion categories

Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ ...

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### Is an integral fusion category pseudo-unitary over a finite field?

Here are two propositions in the book Tensor Categories:
Proposition 9.5.1. A pseudo-unitary fusion category admits a unique
spherical structure.
Proposition 9.6.5. Let $\mathcal{C}$ be ...

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### Is there a fusion category not Grothendieck equivalent to a unitary one?

We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) for the notion of (unitary) fusion category. Two fusion categories are Grothendieck equivalent if they have the ...