18
votes

### Sum of squares and divisibility

This is not a complete answer, but just a way to transform the problem into one that can be attacked by brute force in some known way.
Write $d_i^2=N/n_i$. Then your relation becomes $$\frac{1}{n_1}+ \...

17
votes

Accepted

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

In a nonabelian setting the correct notion of kernel is given by the kernel pair, and the correct notion of cokernel is given by the cokernel pair. For example, in any category, a morphism $f : a \to ...

12
votes

Accepted

### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

The category $Func(C,C)$ is very rarely braided.
It's a bit like asking "when is the endomorphism algebra of a vector space commutative?"
For example, if $C=Vect\oplus Vect$, then $Func(C,C)$ is ...

11
votes

Accepted

### Twists, balances, and ribbons in pivotal braided tensor categories

Question 2: Given a pivotal braided category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with twists under which $\mathcal{C}$ is a rigid balanced category. Conversely, given a rigid ...

11
votes

Accepted

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

I think that there is indeed no such finite group $G$, whether simple or otherwise. Note first that the representation $5_{1}$ can be assumed to be faithful ( for if $K$ is its kernel, then the group $...

11
votes

Accepted

### Local fusion categories

Let $\mathcal{R}$ be a fusion category and $\beta : \mathcal{R} \to \mathrm{Vec}$ an additive monoidal functor.
I first claim that $\beta$ is automatically faithful. (I know why you use "top ...

10
votes

### What are the necessary conditions for a real number to be a cyclotomic integers？

Since you are looking for restrictions on quantum dimensions of objects in unitary fusion categories, you also want your cyclotomic integers to be totally real, as they are Frobenius-Perron ...

10
votes

Accepted

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

Consider the category of functors from the poset $(\mathbb{Q}_{\geq0},\leq)$ to finite-dimensional vector spaces, and let $\mathcal{A}$ be the full subcategory consisting of functors $F$ for which ...

10
votes

Accepted

### Mac Lane strictness theorem and categorifiability of fusion rings

@Todd Trimble is correct; a strict fusion category is just a fusion category that is strict as a monoidal category. (Being strict or being fusion is just a property of a monoidal category that can ...

10
votes

Accepted

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

Any unitary fusion category has a canonical spherical structure. See Example 2.12 of this paper and the references therein. It also follows from a more general result Prop 8.23 in this paper. So you ...

10
votes

Accepted

### Is there a strongly noncommutative fusion category?

Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\...

9
votes

Accepted

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

The answer to your first question is no. The category of modules over a commutative algebra has a tensor category structure such that the forgetful functor is a tensor functor, but the category of ...

9
votes

Accepted

### A cohomology theory for fusion categories

There is no such cohomology theory known (in particular, this is not related to Davydov-Yetter cohomology which is about deformations and vanishes for finite groups). In my mind this is a very ...

9
votes

Accepted

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

Here is a more general statement, see also Lemma 1.2 in [1].
Lemma: Let $Z$ be a self-dual $kG$-module which admits a non-degenerate $G$-invariant symmetric (alternating) bilinear form $b$. Suppose ...

9
votes

Accepted

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

I asked Alexis Virelizier (coauthor of the book mentioned above) by e-mail. Here is his answer (reproduced with his authorisation):
The answer is yes. See Theorem 1.1 (page 2291) in the following ...

8
votes

Accepted

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

An example of MTC is Drinfeld double of a finite group $G$ (over any field of
characteristic zero). This category contains representation category of $G$ as a subcategory. So all endomorphisms rings ...

8
votes

Accepted

### Open questions on (finite) tensor categories

We made a list of open problems at an AIM conference a few years ago.

8
votes

Accepted

### Brauer-Picard for a fusion category coming from a quantum group

As far as I know, no one has written this up, but I think you should be able to find the Brauer-Picard groupoid for quantum groups at roots of unity by the following techniques. Now that I've written ...

8
votes

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

This is really an answer to the question in the comment of Noah's answer.
Let $F$ be a solution to the pentagon equations for some fusion category $\mathcal C$. Pivotal structures on $\mathcal C$ ...

8
votes

Accepted

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

Matrix categories, $\mathrm{End}(\mathrm{Vec}^{\oplus n})$. (The identity on each copy of $\mathrm{Vec}$ are summands of the identity.)
(You can generalize this example by putting fusion categories ...

8
votes

Accepted

### Sum of squares and divisibility

I hope so. But please double check (or, better, simplify) the argument below.
Denote $N=qs^2$ for $q$ squarefree. Then each $d_i$ divides $s$, say $d_i=s/m_i$ and we get $$q=1/s^2+\sum_{i=1}^r 1/m_i^2,...

7
votes

### Why are fusion categories interesting?

(Unitary) fusion categories are interesting in physics because they classify gapped phases on the boundary of 2+1D quantum states of matter. Similarly, unitary modular tensor categories are ...

7
votes

Accepted

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

Let's forget about the monoidal structure for a moment. $C$ is semisimple, and semisimplicity means that to give a functor out of $C$ is the same thing as specifying where it sends simple objects. ...

7
votes

Accepted

### Fusion categories: If infinity were an integer

Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,...

7
votes

Accepted

### What are the necessary conditions for a real number to be a cyclotomic integers？

I suspect you might be looking for the following fact:
If $x$ is a cyclotomic integer, and $p$ a prime does not divide the discriminant, then the minimal polynomial of $x$ factors modulo $p$ into ...

7
votes

### When modular tensor categories are equivalent?

A tensor category includes the information of a tensor product, which is something that takes objects and returns objects. This means that a tensor functor can't just "preserve tensor product" it ...

6
votes

### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

This is in some sense an unnatural question to ask. Endofunctors only form a monoidal category in general, and if you want a braiding, that's not just an extra property: it's extra structure. Where ...

6
votes

Accepted

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

Despite my love of the finite group game, let me give an argument that doesn't use the classification of groups of order 18. The 1-dimensional objects correspond to representations of the ...

6
votes

Accepted

### No basis change in a fusion ring allowed?

That's right, fusion rings have more data than just being a ring, they also have a preferred basis corresponding to the simple objects. So two fusion rings can be isomorphic as rings but not as ...

6
votes

Accepted

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

Short answer: Look for papers on "deequivariantization". (I think the original references are by Müger and Brugières, but I am not sure whether they used the term "deequivariantization".)
Longer ...

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