Does every Frobenius algebra in a monoidal *-category give a Q-system? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:01:49Z http://mathoverflow.net/feeds/question/41210 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41210/does-every-frobenius-algebra-in-a-monoidal-category-give-a-q-system Does every Frobenius algebra in a monoidal *-category give a Q-system? Noah Snyder 2010-10-05T21:51:50Z 2013-05-23T02:27:01Z <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/41210/does-every-frobenius-algebra-in-a-monoidal-category-give-a-q-system/41462#41462 Answer by Noah Snyder for Does every Frobenius algebra in a monoidal *-category give a Q-system? Noah Snyder 2010-10-07T22:05:39Z 2010-11-04T23:28:02Z <p>This answer is wrong, the semion category has a nontrivial associator and so 1+X is not an algebra there. See Tobias's answer. </p> <hr> <p>I think the answer to this question is "no." Below I explain a counterexample.</p> <p>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 <a href="http://arxiv.org/abs/math/0111204" rel="nofollow">From Subfactors to Categories and Topology I</a> which explains that Q-systems are always real.</p> <p>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 <a href="http://arxiv.org/abs/0712.1377" rel="nofollow">On classification of modular tensor categories</a> where they note that the nontrivial object has "Frobenius-Schur indicator -1", in other words it's pseudoreal.</p> http://mathoverflow.net/questions/41210/does-every-frobenius-algebra-in-a-monoidal-category-give-a-q-system/44852#44852 Answer by Tobias Hagge for Does every Frobenius algebra in a monoidal *-category give a Q-system? Tobias Hagge 2010-11-04T18:00:58Z 2010-11-04T18:00:58Z <p>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.</p> http://mathoverflow.net/questions/41210/does-every-frobenius-algebra-in-a-monoidal-category-give-a-q-system/131544#131544 Answer by César Galindo for Does every Frobenius algebra in a monoidal *-category give a Q-system? César Galindo 2013-05-23T02:27:01Z 2013-05-23T02:27:01Z <p>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 <a href="http://lanl.arxiv.org/abs/1105.5048" rel="nofollow">GMR</a>, 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. </p>