# Tagged Questions

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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### When modular tensor categories are equivalent?

I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there. I would like to know when we say that two modular tensor categories are equivalent. Is it ...
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### Mac Lane strictness theorem and categorifiability of fusion rings

The Mac Lane strictness theorem states that any monoidal category is monoidally equivalent to a strict monoidal category (see here section 2.8). Q1: Is it true that any fusion category is monoidally ...
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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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### Is the modularisation of a unitary fusion category always unitary?

Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
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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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### 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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### Fusion categories: If infinity were an integer

Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and ...
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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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### For what $G$ is $Rep(D(S_3))_{ad}$ Grothendieck equivalent to $Rep(G)$?

Given a fusion category $\mathcal C$, the Grothendieck Ring $K_0(\mathcal C)$ is the $\mathbb Z$-based ring whose basis elements correspond to isomorphism classes of simple objects and whose ...
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### Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
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### What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$. ...
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### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...
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### Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes?

Let $C$ be a fusion category with simple objects $X_1,...,X_n$, and let $Y_1,...,Y_n$ be objects with each $Y_i$ isomorphic to $X_i$. Is there a monoidal auto-equivalence $F:C \rightarrow C$ which ...
For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...