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5. Objects in fusion categories

    1. Problem 5.1.

      [Rowell] What values can \dim(X) take in [2,3] for X\in\mathcal{C}, a braided fusion category?
          The dimensions \displaystyle\frac{\sqrt{3}+\sqrt{7}}{2}, \frac{1+\sqrt{13}}{2} do not appear in the braided case.
        1. Remark. [Scott Morrison] These two dimensions come from Cyclotomic integers, fusion categories, and subfactors Frank Calegari, Scott Morrison and Noah Snyder, Communications in Mathematical Physics Volume 303, Issue 3 (2011), pp. 845-896 [arXiv:1004.0665].
            • Problem 5.2.

              [Snyder] What are all the \mathcal{C} generated by X with \text{FPdim}(X)\leq2, and X not self-dual? Are they group theoretical if \dim(X)=2?
                  Can do if X is self-dual and unitary (this is the subfactor case).

              Snyder: I think I can do it if X\otimes X^*\cong X^*\otimes X,

              Rowell: enough if X is self-dual and the Grotheneick ring is commutative.
                • A fusion category version of supertransitivity

                  Problem 5.3.

                  [Jones] In a fusion category, is there an upper bound on the N such that X^{\otimes N} is a simple object (where \dim(X)>1)?
                      Haagerup: If the fish exist, then no.
                    • Problem 5.4.

                      There are accumulation points from below for \text{FPdim}(X) for an object in a fusion category or [M\colon N] for finite depth subfactors. Are there any accumulation points from above?
                          Note that there are no accumulation points at all for \text{FPdim}(\mathcal{C}) for a fusion category \mathcal{C} by Ocneanu rigidity.
                        1. Remark. [Scott Morrison] Are there non-integer accumulation points?

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