# Unitary structures on fusion categories

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.

• 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 $\mathrm{C}^\ast$? – Marcel Bischoff May 26 '13 at 12:01
• 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? – César Galindo May 28 '13 at 18:16
• 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. – Noah Snyder May 30 '13 at 21:54
• 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? – César Galindo May 31 '13 at 20:47
• 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 $X \sqrt{X X^\dagger}^{-1}$? – Noah Snyder Jun 13 '13 at 4:37