# Tag Info

Accepted

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

A characterization of Drinfeld centers of fusion categories is given in this paper as braided fusion categories containing a so-called Lagrangian algebra.

### 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 ...
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 ...
Accepted

### Open questions on (finite) tensor categories

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

### 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 ...
Accepted

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

The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural ...
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 ...

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

If $3 \not \in S$ then the answer to Question($S$) is yes. There are integral commutative fusion rings which are not of Frobenius type. Examples: Non-simple: rank $4$, FPdim $15$, type $[1,1,2,3]$, ...
Accepted

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

Yes, according to Andrew Schopieray. He just provided a categorifiable fusion ring, of rank 6 and multiplicity 2, without pseudounitary categorification (so without unitary categorification), in the ...
Accepted

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

Here is a necessary condition for a group $G$ such that Rep($G$) is Grothendieck equivalent to a modular category: there is a bijection between irreducible complex characters of $G$ and conjugacy ...
Accepted

### Drinfeld center of a Deligne tensor product

By Cororllary 3.26 of arxiv:1009.2117, any braided tensor functor out of a non-degenerately braided fusion category is automatically fully faithful. Since $Z(\mathcal{C})\boxtimes Z(\mathcal{D})$ is ...
Accepted

### Does every enriched functor preserve tensors?

No. The functor "take a vector space to its double dual" is linear but does not preserve tensors with infinite-dimensional vector spaces. Enriched functors are automatically "lax tensored". An ...
Accepted

### Module categories for Fibonacci anyons

There is only one equivalence class of indecomposable module categories, namely the trivial one. Let us look into the possible algebras. They are $1$ and $1\oplus \tau$, and both have a unique ...
### What is a true invariant of $G$-crossed braided fusion categories?
The information in the $\theta$'s is very weak, for example if a FC admits a symmetric structure with $\theta_i=-1$ for at least one $i$, it also admits a symmetric structure with $\theta\equiv 1$. ...