Commutators of relative and unrelative elementary subgroups in Chevalley groups

Abstract

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 $\Phi$ 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 $G(\Phi,R)$ of type $\Phi$ over $R$. The unrelative elementary subgroup $E(\Phi,A)$ of level $A$ is generated (as a group) by the elementary unipotents $x_{\alpha}(a)$, $\alpha\in\Phi$, $a\in A$, of level $A$. Its normal closure in the absolute elementary subgroup $E(\Phi,R)$ is denoted by $E(\Phi,R,A)$ 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 $[E(\Phi,R,A),E(\Phi,R,B)]$, where $A$ and $B$ are two ideals of $R$. It turned out that one can take Stein---Tits---Vaserstein generators of $E(\Phi,R,AB)$, plus elementary commutators of the form $y_{\alpha}(a,b)=[x_{\alpha}(a),x_{-\alpha}(b)]$, 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 $E(\Phi,R,AB)$ the commutators $y_{\alpha}(a,b)$ behave as symbols. We discuss also some further variations and applications of these results.