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Generation of relative commutator subgroups in Chevalley groups. II

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 March 2020

Nikolai Vavilov  and
Zuhong Zhang
Nikolai Vavilov
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, 29 Line 14th (Vasilyevsky Island), 199178St. Petersburg, Russia ([email protected])
Zuhong Zhang
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing10081, China ([email protected])
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Abstract

In the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

Keywords

Chevalley groupselementary subgroupsgeneration of mixed commutator subgroupsstandard commutator formula

MSC classification

Primary: 20G35: Linear algebraic groups over adèles and other rings and schemes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 497 - 511
Copyright
Copyright © Edinburgh Mathematical Society 2020

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