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

## 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. [org.aimpl.user:scott@tqft.net] 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.