AI 中文总结
本文解决了F4型单连通单复代数群对应的量子坐标环𝒪_q(F4)的量子簇代数结构问题，通过统一论证证明其由广义量子子式生成，完善了该领域的相关结论。
AI 中文摘要
设G为单连通单复代数群，Oya、Qin与Yakimov已证明，除F4型外，量子坐标环𝒪_q(G)由广义量子子式生成，因此具有量子簇代数结构。本文通过适用于G2、F4、E8型的统一论证解决F4型情形，核心思路是利用E8型现有证明（依赖量子伴随表示的Lusztig典范基），替换为拟极小晶体的组合结构，最终得出𝒪_q(F4)也具有量子簇代数结构的结论。
英文摘要
Let $G$ be a simply connected simple complex algebraic group. It is proved by Oya, Qin, and Yakimov that the quantized coordinate ring $\mathcal{O}_q(G)$ is generated by generalized quantum minors, and therefore carries a quantized cluster algebra structure, for all $G$ but type $F_4$. In this article, we settle the $F_4$ case by an argument uniform across $G_2$, $F_4$, and $E_8$. The main idea is to bootstrap the existing proof in type $E_8$, which relies on Lusztig's canonical basis of the quantum adjoint representation, and replace it with the crystal combinatorics of the quasi-minuscule representation. As a consequence, we prove that $\mathcal{O}_q(F_4)$ also has a quantized cluster algebra structure.
CommentsSome small changes were made, and some parts were rewritten
