TY - JOUR

T1 - Commutators of relative and unrelative elementary subgroups in Chevalley groups

AU - Vavilov, Nikolai

AU - Zhang, Zuhong

PY - 2020/3/13

Y1 - 2020/3/13

N2 - n 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 gener-ating sets for commutators of relative elementary subgroups in Chevalley groups.Namely, let Φ be a reduced irreducible root system of rank≥2, letRbe a commu-tative ring and letA, Bbe two ideals ofR. We consider subgroups of the ChevalleygroupG(Φ, R) of type Φ overR. The unrelative elementary subgroupE(Φ, A) oflevelAis generated (as a group) by the elementary unipotentsxα(a),α∈Φ,a∈A,of levelA. Its normal closure in the absolute elementary subgroupE(Φ, R) is de-noted byE(Φ, R, A) and is called the relative elementary subgroup of levelA. Themain results of [11,35] consisted in construction of economic generator sets for themutual commutator subgroups [E(Φ, R, A), E(Φ, R, B)], whereAandBare twoideals ofR. It turned out that one can take Stein—Tits—Vaserstein generators ofE(Φ, R, AB), plus elementary commutators of the formyα(a, b) = [xα(a), x−α(b)],wherea∈A,b∈B. Here we improve these results even further, by showing that infact it suffices to engage only elementary commutators corresponding toonelongroot, and that moduloE(Φ, R, AB) the commutatorsyα(a, b) behave as symbols.We discuss also some further variations and applications of these results.

AB - n 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 gener-ating sets for commutators of relative elementary subgroups in Chevalley groups.Namely, let Φ be a reduced irreducible root system of rank≥2, letRbe a commu-tative ring and letA, Bbe two ideals ofR. We consider subgroups of the ChevalleygroupG(Φ, R) of type Φ overR. The unrelative elementary subgroupE(Φ, A) oflevelAis generated (as a group) by the elementary unipotentsxα(a),α∈Φ,a∈A,of levelA. Its normal closure in the absolute elementary subgroupE(Φ, R) is de-noted byE(Φ, R, A) and is called the relative elementary subgroup of levelA. Themain results of [11,35] consisted in construction of economic generator sets for themutual commutator subgroups [E(Φ, R, A), E(Φ, R, B)], whereAandBare twoideals ofR. It turned out that one can take Stein—Tits—Vaserstein generators ofE(Φ, R, AB), plus elementary commutators of the formyα(a, b) = [xα(a), x−α(b)],wherea∈A,b∈B. Here we improve these results even further, by showing that infact it suffices to engage only elementary commutators corresponding toonelongroot, and that moduloE(Φ, R, AB) the commutatorsyα(a, b) behave as symbols.We discuss also some further variations and applications of these results.

KW - группы Шевалле, элементарные подгруппы, корождение смешанных коммутаторов, коммутационные формулы, символы в алгебраической К-теории

KW - Chevalley groups

KW - elementary subgroups

KW - generation of mixed commutator subgroups

KW - standard commutator formulae

UR - https://www.researchgate.net/publication/339971819_Commutators_of_relative_and_unrelative_elementary_subgroups_in_Chevalley_groups

M3 - Article

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

ER -