# Questions tagged [fusion-categories]

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117
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### Is there a noncommutative simple fusion ring?

A fusion ring $\mathcal{F}$ is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying axioms slightly ...

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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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### 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 ...

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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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### 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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### Non weakly-group-theoretical integral fusion category

Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$ below)?
$$\scriptsize{\begin{...

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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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### Extended cyclotomic criterion for unitary categorification

According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object $a$ of a fusion category over $\mathbb{C}$ is a cyclotomic integer. Now, FPdim($a$) is the maximal ...

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### Is there an integral fusion ring which is not of Frobenius type?

Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...

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### The simple unitary fusion categories of multiplicity one

Here are two families of simple unitary fusion categories of multiplicity one:
$Vec(C_p)$ with $C_p$ the cyclic group of order $p$ (one or prime),
The even part of Temperley-Lieb $A_{2n}$ with $n \...

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### A fusion ring identity

Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...

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### Existence of a unitary fusion category with this relation ruled out on finite groups

In this answer, Geoff ruled out the existence of a finite group $G$ such that the fusion category $\mathrm{Rep}(G)$ has simple objects $5_1$ and $7_1$ of FPdim $5$ and $7$ resp., with (for some object ...

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### Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation

In this post, irrep and dim mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) ...

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### What is the smallest rank for a noncommutative fusion ring?

A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying ...

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### Semisimplicity of the tensor identity in a multifusion category over an arbitary field

For a multifusion category $ \mathcal{C} $ over an algebraically closed field it is known that $ \text{End}(\mathcal{1}) $ is a commutative semi-simple algebra. See, for example, Theorem 4.3.1 in [1]. ...

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### Fusion category and induction matrix to its Drinfeld center: combinatorial properties

This question is inspired by this paper of Scott Morrison and Kevin Walker.
Consider a fusion category $\mathcal{C}$ of rank $r$, and its Drinfeld center $Z(\mathcal{C})$ of rank $s$.
Let $N_i = (n_{...

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### Is there a semisimple Hopf algebra Grothendieck equivalent to a strictly weak one?

By Corollary 2.22 in On fusion categories (by Pavel Etingof, Dmitri Nikshych and Viktor Ostrik) any fusion category is equivalent to the category of finite dimensional representations of a semisimple ...

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### How to translate connection on four graphs to quantum 6j symbols

I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...

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### Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?

Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in ...

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### An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see Kashina, Sommerhaeuser, and Zhu - On higher ...

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### Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\...

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### Are there irreducible multi-fusion categories that are not fusion categories?

Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a ...

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### 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," "...

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### How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").
Physically this modular fusion category describes the ...

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### What is a true invariant of $G$-crossed braided fusion categories?

Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences.
(Spherical) fusion categories have ...

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### Is there a “killing” lemma for G-crossed braided fusion categories?

Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.
Premodular categories
In braided ...

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### Morita equivalent algebras in a fusion category

Let $\mathcal{C}$ be a braided $\mathbb{k}$-linear fusion category ($\mathbb{k}$ algebraically closed; if necessary to answer my question you can also assume $\mathcal{C}$ to be pivotal or even ...

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### What do “pivotal” and “spherical” mean for (unitary) fusion categories on the level of the $F$-symbols?

For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one ...

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### Are there two groups which are categorically Morita equivalent but only one of which is simple

Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) ...

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### A cohomology theory for fusion categories

It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^*)$, up to equivalence of categories. ($k^*$ is ...

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### How does the relative Drinfeld center interact with the relative Deligne tensor product?

Let $\mathcal{C}$ be a fusion category, and $\mathcal{M}, \mathcal{N}$ semisimple $(\mathcal{C}, \mathcal{C})$-bimodule categories. The left $\mathcal{C}$-action is denoted as $- \triangleright - \...

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### What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?

Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...

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### 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,
...

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### 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 ...

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### Triviality of Semisimple Hopf Algebras of Cyclic Dimension

A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...

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### Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)

To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post).
A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together ...

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### Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?

A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^kh_k $...

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### Does every enriched functor preserve tensors?

Let $\cal{P}$ be a $k$-linear semisimple abelian rigid monoidal category with finite dimensional (over $k$) Hom-spaces (for a field $k$).
By a tensored $\cal{P}$-category we mean a $\cal{P}$-category ...

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### On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category.
Question: What about the converse, i.e., can we characterize every unitary modular tensor ...

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### Is the Nichols-Richmond theorem true for integral fusion rings?

The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper.
It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here):
Theorem: ...

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### Fusion Classification of $U_q(sl_N)$ Categories

Frohlich and Kerler classify categories with $SU(2)_k$ fusion rules and Kazdhan-Wenzl expand this to $SU(N)_k$ categories. In both cases, unless I am missing something, the classification are ...

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### Module categories for Fibonacci anyons

What are the module categories over the modular tensor category Fib of Fibonacci anyons?
By Ostrik's work, we know these module categories correspond to separable algebras in Fib. I do not believe ...

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### representation of a group and its center

(I asked the following question at StackExchange but received no answer.)
Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...

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### Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...

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### 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 ...

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### Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...

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### How nontrivial can “central extensions of ribbon fusion categories” be?

In a sense, this is a follow up on this question, but one PhD programme later.
Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...