Relative commutator calculus in Chevalley groups

TL;DR

This paper develops a new, more general relative commutator calculus for Chevalley groups, simplifying previous methods and proving a key mixed commutator formula that addresses an open problem.

Contribution

It introduces a simplified, more general relative commutator calculus for Chevalley groups and proves a significant mixed commutator formula resolving an open problem.

Findings

Established a more general relative commutator calculus.

Proved the mixed commutator formula for Chevalley groups.

Answered an open problem in the theory of Chevalley groups.

Abstract

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups $G(\Phi,R)$, $\rk(\Phi)\geq 2$, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in \cite{RZ,RNZ}. As an application we prove the mixed commutator formula, \[ \big [E(\Phi,R,\ma),C(\Phi,R,\mb)\big ]=\big [E(\Phi,R,\ma),E(\Phi,R,\mb)\big], \] for two ideals $\ma,\mb\unlhd R$. This answers a problem posed in a paper by Alexei Stepanov and the second author.