论文标题
不完美字段的还原代数组的亚组的完全降低性iv：$ f_4 $示例
Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example
论文作者
论文摘要
令$ k $为一个不完美的封闭场。令$ g $为一个连接的还原代数集团，定义了$ k $。我们研究了Serre的$ g $的子组完全降低的概念的理性问题。特别是，我们介绍了连接的非亚伯$ k $ -subgroup $ h $ $ g $的第一个例子，这是$ g $ - 可以还原，但不是$ g $ - 在$ k $上可还原，而不是$ k $的第一个示例，也是连接的nonabelian $ k $ k $ k $ -subgroup $ g $ g $ g $ g $ g $ g $ g的第一个示例可还原。这是新的：所有以前已知的此类示例都是针对有限（或非连接）$ h $和$ h'$的。
Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present the first example of a connected nonabelian $k$-subgroup $H$ of $G$ that is $G$-completely reducible but not $G$-completely reducible over $k$, and the first example of a connected nonabelian $k$-subgroup $H'$ of $G$ that is $G$-completely reducible over $k$ but not $G$-completely reducible. This is new: all previously known such examples are for finite (or non-connected) $H$ and $H'$ only.
