
## 2. How many fusion categories...

1. #### Problem 2.1.

[Davydov] Can we find all fusion categories with a given smallest simple object (which is not invertible)?
1. Remark. [Scott Morrison] For example, we know that the smallest possible fusion dimension $1/2(\sqrt{3}+\sqrt{7})$ is realized by the Izumi-Xu-Ostrik fusion category from [arXiv:1004.0665]. What other fusion categories contain an object with this dimension?
• #### Problem 2.2.

[Wang] Is there an effective version of Ocneanu rigidity? Is there a sub-exponential bound on the number of unitary fusion categories with respect to $N$, the sum of all the fusion multiplicities $N_{i,j}^k$?
• #### Problem 2.3.

[Wenzl] How many fusion categories have the same given fusion rules?
We can do this for $SU(N)_k$, assuming the category is braided. It seems you should be able to do this for all quantum groups at roots of unity. You have to look at nice examples, or it is intractable.
• #### Problem 2.4.

What can you say about all fusion categories $\mathcal{C}$ for which $\#\{\dim(X)|X\in\mathcal{C}\}=2?$

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