2020/03/13 by N. A. Vavilov, Vavilov, Nikolai, Zuhong Zhang +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2003.07230
openalex publication_date 2020/03/13 · openalex created_date 2020/03/23 · openalex updated_date 2026/07/28
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 Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let A,B be two ideals of R. We consider subgroups of the Chevalley group G(Φ,R) of type Φ over R. The unrelative elementary subgroup E(Φ,A) of level A is generated (as a group) by the elementary unipotents xα(a), α∈Φ, a∈ A, of level A. Its normal closure in the absolute elementary subgroup E(Φ,R) is denoted by E(Φ,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(Φ,R,A),E(Φ,R,B)], where A and B are two ideals of R. It turned out that one can take Stein---Tits---Vaserstein generators of E(Φ,R,AB), plus elementary commutators of the form yα(a,b)=[xα(a),x-α(b)], where a∈ A, b∈ 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(Φ,R,AB) the commutators yα(a,b) behave as symbols. We discuss also some further variations and applications of these results.