Fusion Categories Associated to Subfactors with Index $3+sqrt{5}$
Fusion Categories Associated to Subfactors with Index $3+sqrt{5}$
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与索引为 $3 sqrt{5}$ 的子因子相关的融合类别
DOI：
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发表时间：
2016
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通讯作者：
Pinhas Grossman
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文献类型：
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作者：
Pinhas Grossman
我们对 Morita 相当于索引 $3+sqrt{5} $ 的子因子的偶数部分的融合类别进行分类，并对这些融合类别进行模块类别。对于融合类别 $mathcal{C} $，它是自对偶 $3^{mathbb{Z}/2mathbb{Z} ime mathbb{Z}/2mathbb{Z} } $ 子因子的偶数部分，我们表明在 $ mathcal{C}$ 上有 $30$ 个简单模块类别； Morita 等价类中没有其他融合类别； Brauer-Picard 集团的订单为 360 美元。通过首先描述 $ mathbb{Z}/3mathbb{Z} $-等变化 $mathcal{C}^{mathbb{Z}/3mathbb{Z} } $（这是 $4442$ 子因子的偶数部分）的布劳尔-皮卡德群群来间接进行证明。我们证明，$mathcal{C}^{mathbb{Z}/3mathbb{Z} } $ 的森田等价类中恰好存在其他三个融合类别，它们都是 $mathcal{C} $ 的 $mathbb{Z}/3mathbb{Z} $ 分级扩展。这些融合类别中的每一个都承认 $20$ 简单模块类别，并且它们的 Brauer-Picard 组是 $mathcal{S}_3 $。我们还表明，$3^{mathbb{Z}/4mathbb{Z} }$ 子因子的偶数部分的 Morita 等价类中恰好有五个融合类别；每个承认 $7$ 简单模块类别；布劳尔-皮卡德群为 $mathbb{Z}/2mathbb{Z} $。
We classify fusion categories which are Morita equivalent to even parts of subfactors with index $3+sqrt{5} $, and module categories over these fusion categories. For the fusion category $mathcal{C} $ which is the even part of the self-dual $3^{mathbb{Z}/2mathbb{Z} imes mathbb{Z}/2mathbb{Z} } $ subfactor, we show that there are $30$ simple module categories over $ mathcal{C}$; there are no other fusion categories in the Morita equivalence class; and the order of the Brauer-Picard group is $360$. The proof proceeds indirectly by first describing the Brauer-Picard groupoid of a $ mathbb{Z}/3mathbb{Z} $-equivariantization $mathcal{C}^{mathbb{Z}/3mathbb{Z} } $ (which is the even part of the $4442$ subfactor). We show that that there are exactly three other fusion categories in the Morita equivalence class of $mathcal{C}^{mathbb{Z}/3mathbb{Z} } $, which are all $ mathbb{Z}/3mathbb{Z} $-graded extensions of $mathcal{C} $. Each of these fusion categories admits $20$ simple module categories, and their Brauer-Picard group is $mathcal{S}_3 $. We also show that there are exactly five fusion categories in the Morita equivalence class of the even parts of the $3^{mathbb{Z}/4mathbb{Z} }$ subfactor; each admits $7$ simple module categories; and the Brauer-Picard group is $mathbb{Z}/2mathbb{Z} $.