关于一个简单导子的迷向群
On the isotropy group of a simple derivation
摘要：
设$R=K[X_1,\dots, X_n]$是特征为零的域$K$上的$n$变量多项式环，$d$是$R$的$K$-导子.如果$d$：$ \text{Aut}(R)_d :=\{ρ\in \text{Aut}_K(R)|\; ρd ρ^{-1}=d\}$，则考虑各向同性组。在他的博士论文中，Baltazar证明了如果$d$是$K[X_1,X_2]$的简单Shamsuddin导子，那么它的各向同性群是平凡的。他还举了一个例子，非简单的推导，其各向同性组是无限的。最近，Mendes和Pan将这一结果推广到了$K[X_1,X_2]$的任意推导，证明了$K[X_1,X_2]$的推导是简单的，当且仅当它的各向同性群是平凡的。本文证明了多项式环$R=K[X_1,\dots, X_n]$的单Shamsuddin导子的迷向群是平凡的.我们还计算了$K[X_1,X_2]$的导子（不一定是单导子）的其他各向同性群，并证明了它们是有限的循环群。
英文摘要：
Let $R=K[X_1,\dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ \text{Aut}(R)_d :=\{ρ\in \text{Aut}_K(R)|\; ρd ρ^{-1}=d\}$. In his doctoral thesis, Baltazar proved that if $d$ is a simple Shamsuddin derivation of $K[X_1,X_2]$, then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan generalized this result to an arbitrary derivation of $K[X_1,X_2]$ proving that a derivation of $K[X_1,X_2]$ is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring $R=K[X_1,\dots, X_n]$ is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of $K[X_1,X_2]$ and prove that they are finite cyclic groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
