
## 1. General fusion category questions

1. #### Problem 1.1.

Is any integral fusion category unitarizable?
1. Remark. [Cesar Galindo, Eric Rowell] By a recent paper of Galindo, Hong and Rowell, a slightly stronger statement can be proved for weakly group-theoretical categories: they are "completely unitary." Since weakly g.-t. categories conjecturally contain all weakly integral fusion categories, the answer is probably "yes."
• #### Problem 1.2.

Is every integral fusion category weakly group theoretical?
• #### Problem 1.3.

Does pseudo-unitary imply unitarizable?
1. Remark. [Wang] Physicists are really interested in this question. Given a conformal field theory which is not unitary, there is a negative dimension.
• #### Problem 1.4.

Are all fusion categories pivotal? Are all fusion categories spherical? Does it depend on the ground field ($k$ v.s. $\C$)?
• ### "Kaplansky’s sixth conjecture" for fusion categories

#### Problem 1.5.

Is $\displaystyle \frac{\text{FPdim}(\mathcal{C})}{\text{FPdim}(X)}$ an algebraic integer for every $X\in\text{Irr}(\mathcal{C})$?

Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.