Relative commutator calculus in Chevalley groups

JOURNAL OF ALGEBRA

卷:385

Article

Hazrat, Roozbeh¹  Vavilov, Nikolai²  Zhang, Zuhong³ 

[1] Univ Western Sydney, Sch Comp Engn & Math, Penrith, NSW 1797, Australia

[2] St Petersburg State Univ, Dept Math & Mech, St Petersburg, Russia

[3] Beijing Inst Technol, Dept Math, Beijing 100081, Peoples R China

关键词: Chevalley groups;    Elementary subgroups;    Localisation-completion;    Quillen-Suslin lemma;    Commutator formulae;   

DOI  :  10.1016/j.jalgebra.2013.03.011

来源: Elsevier

【 摘 要 】

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) >= 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 Hazrat, Zhang (2011) [34], Hazrat et al. (2011) [33]. As an application we prove the mixed commutator formula, [E(Phi, R, a), G(Phi, R, b)] = [E(Phi, R, a), E(Phi, R, b)], for two ideals a, b (sic) R. This answers a problem posed in a paper by Alexei Stepanov and the second author. Crown Copyright (C) 2013 Published by Elsevier Inc. All rights reserved.

【 授权许可】

Free