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

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

2010-10-05T21:51:50Z

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.

Answer by Noah Snyder for Does every Frobenius algebra in a monoidal *-category give a Q-system?

2010-10-07T22:05:39Z

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.

Answer by Tobias Hagge for Does every Frobenius algebra in a monoidal *-category give a Q-system?

2010-11-04T18:00:58Z

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.

Answer by César Galindo for Does every Frobenius algebra in a monoidal *-category give a Q-system?

2013-05-23T02:27:01Z

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.