18
$\begingroup$

A unitary fusion category is a fusion category with a $C^*$-tensor structure. Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more than one (non equivalent) unitary structures.

Unitarity is equivalent to the existence of basis such that the $F$-matrices are unitary. This property is to be unitarizible. The question in matricial terms is: suppose that you have two different basis such that the $F$-matrices are unitary, Can be unitary gauged one basis in the other? or in oher words, there exist always unitary change of basis from one basis to the other basis?

[added by A. Henriques]
I see in the comments that there is some confusion in the interpretation of the question, so let me attempt a reformulation:

Is every equivalence $F:\mathcal C\to \mathcal D$ between unitary fusion categories naturally equivalent to a unitary equivalence (i.e., one which is a dagger functor)

I see that Galindo-Hong-Rowell have some results in that direction (see $\S$5.5, on page 21 of his paper), but they don't exactly addresses this question.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ I am curious about this, too. I guess there are also fusion categories without a $\mathrm{C}^\ast$-tensor structure, what goes wrong there? Do you happen to have a reference where the notion of a "unitary fusion category" is used (or even better) introduced? How would I call a MTC which is $\mathrm{C}^\ast$? $\endgroup$
    – Marcel Bischoff
    May 26, 2013 at 12:01
  • 1
    $\begingroup$ There are many places with the definition of unitary fusion category, for example lanl.arxiv.org/abs/1209.2022 The MTC with a C^*-tensor structure are called unitary MTC. An example of a non-unitary fusion category is the Yang-Lee category. The problem is that every unitary fusion category is pseudo-unitary in the sense of arxiv.org/abs/math/0203060. In fact, an other natural question (I guess it in ENO): Is every pseudo-unitary fusion category a unitary fusion category? $\endgroup$
    – César Galindo
    May 28, 2013 at 18:16
  • 2
    $\begingroup$ I'm not totally sure, but I think that if a unitary structure exists then it's unique. This is because unitarity should be equivalent to saying that the F-matrices can be gauged to become unitary matrices, and that's a condition not a structure. $\endgroup$
    – Noah Snyder
    May 30, 2013 at 21:54
  • 1
    $\begingroup$ Yes, you are right, unitarity is equivalent to the existence of a basis such that the F-matrices are unitary. This property is to be unitarizible. The question in matricial terms is: suppose that you have two different basis such that the F-matrices are unitary, Can be unitary gauged one basis in the other? or in oher words, there exist always a unitary change of basis from one basis to the other basis? $\endgroup$
    – César Galindo
    May 31, 2013 at 20:47
  • 1
    $\begingroup$ But isn't that just polar decomposition? That is to say, if you can conjugate one unitary by $X$ to get another, then you can also get between them by conjugating by the unitary matrix $X \sqrt{X X^\dagger}^{-1}$? $\endgroup$
    – Noah Snyder
    Jun 13, 2013 at 4:37

1 Answer 1

Reset to default
2
$\begingroup$

Reutter's recent paper "uniqueness of unitary structure for unitarizable fusion categories" answers your question in the affirmative (link: https://arxiv.org/pdf/1906.09710.pdf).

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.