Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 coproduct on A' is an isometry)? In other words, wlog can one assume that the coproduct is the * of the product?

The motivation of this question is to understand whether the theory of irreducible Frobenius algebra objects in monoidal *-categories actually agrees on the nose with the theory of irreducible subfactors or whether there's a small loophole.

share|improve this question
    
So here's a sketch of how I was thinking of proving this before I thought I had a counterexample. Look at the 2-category of 1-1, 1-A, A-1, and A-A bimodules. We want to prove that this is a 2-C*-category and then by work of Longo we could conclude that A is a Q-system. But all those bimodule categories have a forgetful functor to the original category, so you just use the original *-structure to inherit a *-structure on the 2-category. You then check that this is a C* structure and you're done. –  Noah Snyder Nov 4 '10 at 22:30
add comment

3 Answers 3

This is a comment on Noah's answer, posted as an answer due to lack of reputation. The semion MTC is inequivalent to Vec(Z/2) as a fusion category; it is the other rank two fusion category. Confusingly, there is a change in sign in one of the F-matrices AND a change in sign in the pivotal structure which gives unitarity; the two occur simultaneously in most diagrams.

share|improve this answer
    
Ah very good. Right, I think at one point a long time ago I worked this out and then I forgot. There are 4 *-fusion categories here, you can move between them by changing * structure or by changing associator. If you only do one or the other then you negate dimensions, but if you do both you keep dimensions the same. So only 2 of them are positive definite: the usual associator and usual *-structure or unusual associator and unusual *-structure. In particular, 1+X is only an algebra object in the former case (because the module category is Vec giving a fiber functor). –  Noah Snyder Nov 4 '10 at 21:45
add comment

This is not a full answer. The answer is yes for weakly group-theoretical fusion categories. The question is equivalent to: let C be a unitary fusion category, does every indecomposable C-module category admit a compatible unitary structure (see GMR, for all definitions). In Theorem 5.20, we prove that a weakly group-theoretical fusion category admits a unique unitary structure and every indecomposable module category also admits a unique compatible unitary structure.

share|improve this answer
add comment

This answer is wrong, the semion category has a nontrivial associator and so 1+X is not an algebra there. See Tobias's answer.


I think the answer to this question is "no." Below I explain a counterexample.

Consider the fusion category Vec(Z/2) with two objects 1 and X. The object 1+X has a natural Frobenius algebra structure (just think of it as the group ring C[Z/2]). However, Vec(Z/2) has two different *-structures: the usual *-structure where X is real (aka orthogonal) and the *-structure where X is pseudoreal (aka quaternionic aka symplectic). In the latter case 1+X can't have a Q-system structure by remark 3 on page 30 of Mueger's From Subfactors to Categories and Topology I which explains that Q-systems are always real.

The tricky point in the above is checking that Vec(Z/2) really does have a second *-structure. I worked this out diagrammatically, but it can also be realized by looking atU_q(sl_2) when q is a primitive 6th root of unity since spin 1/2 reps are pseudoreal. This tensor *-category is called the called the "semion" theory in Section 5.3.1. of Rowell-Strong-Wang's On classification of modular tensor categories where they note that the nontrivial object has "Frobenius-Schur indicator -1", in other words it's pseudoreal.

share|improve this answer
    
With this second -structure, does T*T >= 0 hold for any morphism? If not, your counter-example would be ruled out for C-categories. –  pasquale zito Oct 7 '10 at 22:34
    
Yes, I think T*T>=0, unless I somehow made a mistake. In particular in the Rowell-Strong-Wang paper they're only considering unitary modular tensor categories, and unitary means C*. –  Noah Snyder Oct 7 '10 at 22:49
    
Looks like Tobias is right above. –  Noah Snyder Nov 4 '10 at 21:42
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.