# Questions tagged [fusion-categories]

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40
questions with no upvoted or accepted answers

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### 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 ...

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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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### 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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### 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 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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### 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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### 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}, ..., d_{r}...

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### 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 $...

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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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### 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 ...

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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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### Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...

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### Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial.
proof: First if $...

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### 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 d(...

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### 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 ...

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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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### 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 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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### 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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### 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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### Involution of unital based ring (Grothendieck ring of a fusion category)

Let $A$ be a unital based ring in the sense of [Ostrik, arXiv:math/0111139]. As part of the data we have a base $B = \{b_i\}_{i\in I}$, and an involution $i \mapsto \bar i$ of $I$ whose induced map $\...

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### 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 \hat{\mathbb{...

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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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### 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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### 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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### 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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### 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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### 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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### Lluís Puig unpublished notes

I have been reading Linckelman Notes and the Broto-Levi-Oliver articles about fusion systems and they always make mention about the Lluís Puig notes that he gave years ago, so I'm asking if anyone has ...

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### 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 ...

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### 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 ...

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### “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 http://arxiv.org/abs/hep-th/...

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### 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$.
...

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### 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 $\mathcal{R}_{1}$ ...

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### 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 $SU(2)$....

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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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### 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 $N$...

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### 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 ...

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### 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 ...