Questions tagged [fusion-categories]
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172 questions
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Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
5
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1
answer
178
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Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
3
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Can fusion categories have an adjoint?
The question is a bit provocative, and in reality I mean the following:
Some fusion categories $F$ are generated by taking the irreps $R$ of a quantum Lie algebra at some root of unity (call this ...
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2
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1
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78
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Does there exist a nontrivial triangular weak Hopf algebra?
Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair
($H,\mathcal{R}$) where $H$ is a WHA and
$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...
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Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
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4
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295
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Is there an integral fusion category of the Ising type?
In [EGNO, Section 8.27.3], we read:
Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropic
category (namely, the core of ${\mathcal C}$) using finite groups (via the
...
4
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0
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227
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Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?
In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the ...
3
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0
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98
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Isomorphic objects have the same dimension (pivotal categories)
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
- 143
3
votes
2
answers
135
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Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?
Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...
- 727
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Some fusion rings/categories I don't recognize
Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
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8
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1
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Software for working with fusion categories
One way to describe fusion categories is via a fusion system: several lists of numbers that define the fusion ring, associator, braiding (if it exists), etc. Often, these sets of numbers are quite big,...
- 293
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0
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326
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Are there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
7
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258
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Does a fusion ring with F and R-symbols uniquely determine a braided tensor category?
Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such ...
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5
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1
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349
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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 $\...
- 173
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128
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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 ...
- 293
2
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1
answer
110
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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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3
votes
1
answer
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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 ...
- 293
8
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1
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309
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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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300
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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 ...
2
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0
answers
102
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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 ...
- 301
3
votes
2
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233
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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 $\...
5
votes
2
answers
448
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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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504
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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 ...
3
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0
answers
113
views
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....
3
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0
answers
276
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Is there a non-pointed simple integral modular fusion category?
The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds:
Statement 1 (open): There is a ...
6
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0
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128
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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 (...
3
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0
answers
109
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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 $X$...
4
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246
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Vertex operator algebras and modular fusion categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every ...
1
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0
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106
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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,
$T$ ...
6
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0
answers
148
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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 ...
6
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1
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357
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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 ...
5
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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 ...
2
votes
1
answer
216
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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 ...
3
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1
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492
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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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5
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1
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128
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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-...
4
votes
1
answer
298
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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 (...
2
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0
answers
66
views
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-) ...
- 4,829
3
votes
1
answer
217
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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 ...
26
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3
answers
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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): ...
2
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0
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178
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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 ...
6
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2
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334
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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}(...
- 1,001
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1
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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 $(...
4
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0
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320
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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 ...
7
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1
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335
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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 ...
3
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1
answer
159
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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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1
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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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106
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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 ...
3
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0
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134
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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 ...
5
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0
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135
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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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