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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.