The goal of this paper is to classify fusion categories \({\mathcal {C}}\) which are \(\otimes \)-generated by an object X of Frobenius–Perron dimension less than 2, with the additional mild assumption that the adjoint subcategory of \({\mathcal {C}}\) is \(\otimes \)-generated by the object \(X\otimes X^*\). 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 et al. (Quantum Topol 1(3);209–273, 2010. https://doi.org/10.4171/QT/6), 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 Edie-Michell (Int. J. Math. 29(5):1850036, 2018. https://doi.org/10.1142/S0129167X18500362) 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 \({\mathcal {C}}( \mathfrak {sl}_2, 16)\boxtimes {\mathcal {C}}( \mathfrak {sl}_3,6)\).

中文翻译：

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对融合类别$$ \ otimes $$⊗进行分类-由小Frobenius–Perron维的对象生成

本文的目的是对由Frobenius–Perron尺寸小于2的对象X生成的融合类别\（{\ mathcal {C}} \\）进行分类，\（\ otimes \）\（{\ mathcal {C}} \）的伴随子类别是\（\ otimes \） -由对象\（X \ otimes X ^ * \）生成。由于Morrison和Snyder的结果，最近可以访问此分类，这表明任何此类类别都必须是伴随ADE类别的循环扩展类型。我们在此分类中的主要工具是Etingof等人的结果。（Quantum Topol 1（3）； 209-273，2010年。https：//doi.org/10.4171/QT/6），根据从Brauer-Picard组和Drinfeld计算得出的数据对给定类别的循环扩展进行分类该类别的中心以及Edie-Michell的结果（Int。J. Math。29（5）：1850036，2018. https://doi.org/10.1142/S0129167X18500362）计算Brauer–Picard组和Drinfeld中心伴随ADE类型的类别。我们的分类包括由循环组和ADE类别构成的预期类别类型。更有趣的是，我们在分类中具有类别，这些类别是这些预期类别的非平凡去反变异。所有的最有趣的，我们的分类包括从特殊的量子亚组构建了三个无限族\（{\ mathcal {E}} _ 4 \）的\（{\ mathcal {C}}（\ mathfrak {SL} _4，4）\ ），以及\（{\ mathcal {E}} _ {16,6} \）的\（{\ mathcal {C}}（\ mathfrak {SL} _2，16）\ boxtimes {\ mathcal {C}}（ \ mathfrak {sl} _3,6）\）。