标题： $2$-表示无限代数来自$\operatorname{SL}_3$的非阿贝尔子群。 第二部分：中心扩张和例外情况 显示英文标题

标题： $2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part II: Central extensions and exceptionals

摘要： 设$G \leq \operatorname{SL}_3(\mathbb{C})$为一个非平凡的有限群，作用于$R = \mathbb{C}[x_1, x_2, x_3]$。 我们继续对 arXiv:2505.10683 [math.RT] 中的研究进行探讨，研究所得的斜群代数$R \ast G$何时为一个$3$-预投射代数，该代数属于一个$2$-表示无限代数。 我们考虑从$\operatorname{GL}_2(\mathbb{C}) \hookrightarrow \operatorname{SL}_3(\mathbb{C})$得到的子群，称为类型 (B)，以及例外子群，称为类型 (E) -- (L)。 对于类型(B)的群，我们证明在$R \ast G$上存在一个$3$-预投射结构当且仅当$G$不是$\operatorname{SL}_2(\mathbb{C})$或$\operatorname{PSL}_2(\mathbb{C})$的子群。 对于剩余类型(E) -- (L)的群$G$，每个$R \ast G$都承认一个$3$-预投射结构，除了类型(H)和(I)。 为了证明类型(B)的结果，我们探讨了同源性概念如何与McKay箭图的形状相互作用。 我们详细计算了McKay箭图，使用一种编织风格的启发式方法。 对于例外子群，我们直接计算了McKay箭图以及切割，并讨论了如何以算法方式完成此任务。 这提供了许多新的$2$-表示无限代数的例子，并且与arXiv:2401.10720 [math.RT]，arXiv:2505.10683 [math.RT]一起完成了对$\operatorname{SL}_3(\mathbb{C})$的有限子群的分类，其中$R \ast G$是一个$3$-预投射代数。 显示英文摘要

摘要： Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. We continue our investigation from arXiv:2505.10683 [math.RT] into when the resulting skew-group algebra $R \ast G$ is a $3$-preprojective algebra of a $2$-representation infinite algebra. We consider the subgroups arising from $\operatorname{GL}_2(\mathbb{C}) \hookrightarrow \operatorname{SL}_3(\mathbb{C})$, called type (B), as well as the exceptional subgroups, called types (E) -- (L). For groups of type (B), we show that a $3$-preprojective structure exists on $R \ast G$ if and only if $G$ is not isomorphic to a subgroup of $\operatorname{SL}_2(\mathbb{C})$ or $\operatorname{PSL}_2(\mathbb{C})$. For groups $G$ of the remaining types (E) -- (L), every $R \ast G$ admits a $3$-preprojective structure, except for type (H) and (I). To prove our results for type (B), we explore how the notion of isoclinism interacts with the shape of McKay quivers. We compute the McKay quivers in detail, using a knitting-style heuristic. For the exceptional subgroups, we compute the McKay quivers directly, as well as cuts, and we discuss how this task can be done algorithmically. This provides many new examples of $2$-representation infinite algebras, and together with arXiv:2401.10720 [math.RT], arXiv:2505.10683 [math.RT] completes the classification of finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ for which $R \ast G$ is a $3$-preprojective algebra.