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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.

edited Nov 4 2010 at 22:04

asked Oct 5 2010 at 21:51

Noah Snyder
11.3k●1●26●70

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 2010 at 22:30

Noah Snyder · Answer 1 · 2010-11-04 23:28:02Z UTC

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.

Tobias Hagge · Answer 2 · 2010-11-04 18:00:58Z UTC

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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.

answered Nov 4 2010 at 18:00

Tobias Hagge
81●3

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 2010 at 21:45

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

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