Generation of relative commutator subgroups in Chevalley groups. II

Abstract

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 x α (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 ).

Locations

• Proceedings of the Edinburgh Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View