论文标题
$ f $ -Symbols用于透明haagerup-izumi类别，带有$ g = \ mathbb {z} _ {2n+1} $
The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \mathbb{Z}_{2n+1}$
论文作者
论文摘要
如果涉及任何可逆对象的关联是身份图，则称为融合类别。对于Haagerup-izumi融合圈，带有$ g = \ Mathbb {z} _ {2n+1} $（$ \ Mathbb {z} _3 $ case是与六个简单对象的haagerup融合环），固定的ansatz降低了独立$ f $ f $ - symbols $ s的数量{o \ symbols的数量{ $ \ mathcal {o}（n^2）$，实际上可以解决五角大楼的身份。因此，透明的haagerup-izumi融合类别进行了建设性地分类至$ g = \ mathbb {z} _9 $，将所有已知的haagerup-izumi Fusion类别恢复到此订单，并生产新的订单。透明的haagerup-izumi融合类别还满足$ s_4 $四面体不变性，最多可分类为$ g = \ m atmathbb {z} _ {15} $，以及明显的$ f $ f $ f $ -symbols，包括haagerup $ $ \ mathcalcal cancal cancal cancal {还介绍了Haagerup $ \ Mathcal {H} _2 $ Fusion类别的$ f $ -Symbols。透明的安萨兹（Ansatz）超越了，为新的融合戒指构建新颖的融合类别提供了可行的课程。
A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$ (the $\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\mathcal{O}(n^6)$ to $\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying $S_4$ tetrahedral invariance are further classified up to $G = \mathbb{Z}_{15}$, and the explicit $F$-symbols for the unitary ones, including the Haagerup $\mathcal{H}_3$ fusion category, are compactly presented. The $F$-symbols for the Haagerup $\mathcal{H}_2$ fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.
