2012/12/21 by Roozbeh Hazrat, Hazrat, Roozbeh, N. A. Vavilov +4
Mathematics · #20G35 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:20G35
paper · pdf · doi:10.48550/arxiv.1212.5432
arxiv created 2012/12/21 · openalex publication_date 2012/12/21 · arxiv updated 2012/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let x\a(ξ), \a∈Φ, ξ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by x\a(ξ), \a∈Φ, ξ∈ I. Classical results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as a group E(Φ,R,I) is generated by z\a(ξ,η), where \a∈Φ, ξ∈ I, η∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: i) [xα(ξ),zα(ζ,η)], ii) [xα(ξ),x-α(ζ)], and iii) xα(ξζ), where α∈Φ, ξ∈ I, ζ∈ J, η∈ R. As a group, the generators are essentially the same, only that type iii) should be enlarged to iv) zα(ξζ,η). For classical groups, these results, with much more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results, namely in the recent work of Alexei Stepanov on relative commutator width.