Classifying fusion categories $\otimes$-generated by an object of small Frobenius-Perron dimension

Abstract

The goal of this paper is to classify fusion categories $\otimes$-generated by a $K$-normal object (defined in this paper) of Frobenius-Perron dimension less than 2. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint $ADE$ type. Our main tools in this classification are the results of Etingof, Ostrik, and Nikshych, classifying cyclic extensions of a given category in terms of data computed from the Brauer-Picard group, and Drinfeld centre of that category, and the results of the author, which compute the Brauer-Picard group and Drinfeld centres of the categories of adjoint $ADE$ type. Our classification includes the expected categories, constructed from cyclic groups and the categories of $ADE$ type. More interestingly we have categories in our classification that are non-trivial de-equivariantizations of these expected categories. Most interesting of all, our classification includes three infinite families constructed from the exceptional quantum subgroups $\mathcal{E}_4$ of $\mathcal{C}( \mathfrak{sl}_4, 4)$, and $\mathcal{E}_{16,6}$ of $\mathcal{C}( \mathfrak{sl}_2, 16)\boxtimes \mathcal{C}( \mathfrak{sl}_3,6)$.