AI 中文总结
本文对秩为6的ℤ/2ℤ×ℤ/2ℤ二次融合范畴开展部分分类，明确其格罗滕迪克环的可能范围，并证明其中一类融合环可由特定近群融合范畴的等变约化得到。
AI 中文摘要
若融合范畴𝒞的可逆对象构成的群G(𝒞)同构于ℤ/2ℤ×ℤ/2ℤ，且其余单对象在G(𝒞)的作用下形成一个轨道，则称𝒞为ℤ/2ℤ×ℤ/2ℤ二次融合范畴。本文对秩为6的ℤ/2ℤ×ℤ/2ℤ二次融合范畴进行部分分类。更确切地说，当融合规则重数小于20时，证明其格罗滕迪克环𝒦₀(𝒞)必为9个融合环中的一个，且其中5个的范畴化已为已知结果；证明剩余4个融合环中的1个可实现为ℤ/2ℤ×ℤ/4ℤ+8型近群融合范畴的等变约化形式。
英文摘要
A fusion category $\mathcal{C}$ is said to be $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic if the group $G(\mathcal{C})$ of invertible objects is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and the remaining simple objects form an orbit under the action of $G(\mathcal{C})$. In this paper, we give a partial classification of $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic fusion categories of rank six. More precisely, we show that its Grothendieck ring $\mathcal{K}_0(\mathcal{C})$ must be one of nine fusion rings if the fusion rule multiplicities are less than $20$, and the categorifications of five of them are previously known. We prove that one of the last four fusion rings can be realized as de-equivariantization of a near-group fusion category of type $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}+8$.
Comments21 pages; comments are welcome
