Questions tagged [fusion-categories]

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7
votes
1answer
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
votes
2answers
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
1answer
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
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
2answers
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
votes
1answer
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
2answers
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
1answer
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
8answers
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
2answers
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
votes
4answers
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
votes
6answers
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 ...

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