# Questions tagged [fusion-categories]

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117
questions

**7**

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**1**answer

453 views

### Fusion category and Hopf algebra

Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed ...

**5**

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**2**answers

178 views

### Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...

**7**

votes

**1**answer

303 views

### If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?

If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...

**13**

votes

**1**answer

511 views

### Are there interesting semisimple algebras in non-semisimple categories?

Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple ...

**8**

votes

**1**answer

239 views

### Is there a fusion category with an object which does not commute with its dual?

Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)?
Feel free to add adjectives such as pivotal, spherical, ...

**13**

votes

**3**answers

631 views

### Does every Frobenius algebra in a monoidal *-category give a Q-system?

Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...

**25**

votes

**1**answer

2k views

### Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...

**5**

votes

**1**answer

430 views

### What is Out(G-mod) for a finite group G?

Following the notation of Etingof-Nikshych-Ostrik what is Out(G-mod) for a finite group G?
That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the ...

**4**

votes

**1**answer

504 views

### De-equivariantization by Rep(G)

I'm trying to understand Proposition 2.9 of this paper on weakly group theoretical fusion categories.
First of all I have a problem with understanding the settings for de-equivariantization process. ...

**8**

votes

**2**answers

390 views

### Is there a source for a diagrammatic description of the induction functor C->Z(C)?

Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-...

**7**

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**1**answer

302 views

### Suppose C and D are Morita equivalent fusion categories, can you say anything about R I: C->Z(C)=Z(D)->D?

If C and D are (higher) Morita equivalent fusion categories, then the Drinfel'd centers Z(C) and Z(D) are braided equivalent. Given any fusion category C we have a restriction functor Z(C)->C (by ...

**4**

votes

**2**answers

304 views

### Can “premodular” be relaxed as a condition for uniqueness of Bruguieres/Mueger modularization?

Suppose that C is a ribbon monoidal category with dominant ribbon functors F_1: C->D_1 and F_2: C->D_2 such that D_1 and D_2 are modular tensor categories, does it follow that D_1 and D_2 are ...

**2**

votes

**1**answer

537 views

### Module categories over $Rep(G)$.

Related to this question I also had some troubles to understand the classification of module categories over $Rep(G)$. Specifically, on page 12 of Ostrik's paper what is the category $\mathrm{Rep}^1(\...

**34**

votes

**8**answers

7k views

### 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," "...

**9**

votes

**2**answers

328 views

### Is there a subfactor construction involving 2-groups?

I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an ...

**26**

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**4**answers

2k views

### 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) ...

**13**

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**6**answers

1k views

### How do I describe a fusion category given a subfactor?

I felt like following up on Kate's question. There were some good motivational answers there.
Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M ...