A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is simple.

Question: What are the known examples of strongly simple fusion categories?

Note that if $G$ is a finite group then $Rep(G)$ is simple iff $G$ is simple, but it is strongly simple iff $G$ is cyclic of prime order, because $Rep(G)$ is Morita equivalent to $Vec(G)$ whose fusion subcategories (up to equiv.) are $Vec(H)$ with $H<G$.

The Extended Haagerup fusion categories are strongly simple, see this paper (by Grossman-Morrison-Penneys-Peters-Snyder) classifying their Morita equivalence class. I wrote here the explicit fusion matrices (it is easy to see that they are all simple).



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.