论文标题
Chevalley组中相对和不依赖基本亚组的换向因子
Commutators of relative and unrelative elementary subgroups in Chevalley groups
论文作者
论文摘要
在本文中，这是我们论文的直接续集[10,11,35]与Roozbeh hazrat的关节，我们进一步急剧减少了雪佛兰组相对基本亚组的换向因子的生成集。也就是说，让$φ$是降低的等级$ \ ge 2 $的不可约根系统，让$ r $是一个通勤戒指，让$ a，b $是$ r $的两个理想。我们考虑$ r $ $ $φ$的chevalley $ g（φ，r）的子组。基本UniPOTENTS $x_α（a）$，$α\inφ$，$ a \ in a $ a $ a $ a $ a $，$x_α（a）$，$x_α（a）$x_α（a）$x_α（a）$x_α（a）$x_α（a）$x_α（a）$x_α（a）$ a $ a $ a $ a $的不友存基本亚组$ e（φ，a）$。它在绝对基本亚组$ e（φ，r）$中的正常闭合用$ e（φ，r，a）$表示，称为级别$ a $的相对基本亚组。 [11,35]的主要结果在于为共同换向器子组的经济发电机组构建$ [e（φ，r，a），e（φ，r，b）] $，其中$ a $ a $和$ b $是$ r $的两个理想。事实证明，一个人可以服用$ e（φ，r，ab）$的Vaserstein发电机，加上$y_α（a，b）= [x_α（a），x _ { - α}（α}（b）] $的基本换向器，其中$，$ a \ in $ a \ in $，$ b \ in b $。在这里，我们进一步改善了这些结果，表明实际上只能与{\ IT \/}相对应的长根相关的基本换向因子，而换向器$y__α（a，b）$ beh the Modulo $ e（φ，r，ab）$。我们还讨论了这些结果的进一步变化和应用。
In the present paper, which is a direct sequel of our papers [10,11,35] joint with Roozbeh Hazrat, we achieve a further dramatic reduction of the generating sets for commutators of relative elementary subgroups in Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $A,B$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelative elementary subgroup $E(Φ,A)$ of level $A$ is generated (as a group) by the elementary unipotents $x_α(a)$, $α\inΦ$, $a\in A$, of level $A$. Its normal closure in the absolute elementary subgroup $E(Φ,R)$ is denoted by $E(Φ,R,A)$ and is called the relative elementary subgroup of level $A$. The main results of [11,35] consisted in construction of economic generator sets for the mutual commutator subgroups $[E(Φ,R,A),E(Φ,R,B)]$, where $A$ and $B$ are two ideals of $R$. It turned out that one can take Stein---Tits---Vaserstein generators of $E(Φ,R,AB)$, plus elementary commutators of the form $y_α(a,b)=[x_α(a),x_{-α}(b)]$, where $a\in A$, $b\in B$. Here we improve these results even further, by showing that in fact it suffices to engage only elementary commutators corresponding to {\it one\/} long root, and that modulo $E(Φ,R,AB)$ the commutators $y_α(a,b)$ behave as symbols. We discuss also some further variations and applications of these results.
