Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T08:58:25.116Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (nikolai-vavilov@yandex.ru)
Zuhong Zhang
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing10081, China (zuhong@hotmail.com)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Apte, H. and Stepanov, A., Local-global principle for congruence subgroups of Chevalley groups, Cent. Eur. J. Math. 12(6) (2014), 801–812.Google Scholar
2.Bass, H., K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. 22 (1964), 5–60.CrossRefGoogle Scholar
3.Carter, R. W., Simple groups of Lie type, (Wiley, London, 1972).Google Scholar
4.Hazrat, R. and Vavilov, N., K 1 of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), 99–116.CrossRefGoogle Scholar
5.Hazrat, R. and Vavilov, N., Bak's work on the K-theory of rings, with an appendix by Max Karoubi, J. K-Theory 4 (2009), 1–65.CrossRefGoogle Scholar
6.Hazrat, R. and Zhang, Z., Generalized commutator formulas, Comm. in Algebra 39 (2011), 1441–1454.CrossRefGoogle Scholar
7.Hazrat, R. and Zhang, Z., Multiple commutator formula, Israel J. Math. 195 (2013), 481–505.CrossRefGoogle Scholar
8.Hazrat, R., Petrov, V. and Vavilov, N., Relative subgroups in Chevalley groups, J. K-theory 5 (2010), 603–618.CrossRefGoogle Scholar
9.Hazrat, R., Vavilov, N. and Zhang, Z., Relative unitary commutator calculus and applications, J. Algebra 343 (2011), 107–137.CrossRefGoogle Scholar
10.Hazrat, R., Stepanov, A., Vavilov, N. and Zhang, Z., The yoga of commutators, J. Math. Sci. 179(6) (2011), 662–678.CrossRefGoogle Scholar
11.Hazrat, R., Stepanov, A., Vavilov, N. and Zhang, Z., Commutator width in Chevalley groups, Note Mat. 33(1) (2013), 139–170.Google Scholar
12.Hazrat, R., Vavilov, N. and Zhang, Z., Relative commutator calculus in Chevalley groups, J. Algebra 385 (2013), 262–293.CrossRefGoogle Scholar
13.Hazrat, R., Stepanov, A., Vavilov, N. and Zhang, Z., The yoga of commutators, further applications, J. Math. Sci. 200(6) (2014), 742–768.CrossRefGoogle Scholar
14.Hazrat, R., Vavilov, N. and Zhang, Z., Generation of relative commutator subgroups in Chevalley groups, Proc. Edinb. Math. Soc. 59 (2016), 393–410.CrossRefGoogle Scholar
15.Hazrat, R., Vavilov, N. and Zhang, Z., Multiple commutator formulas for unitary groups, Israel J. Math. 219(1) (2017), 287–330.CrossRefGoogle Scholar
16.Hazrat, R., Vavilov, N. and Zhang, Z., The commutators of classical groups, J. Math. Sci. 222(4) (2017), 466–515.CrossRefGoogle Scholar
17.Mason, A. W., A note on subgroups of GL(n, A) which are generated by commutators, J. Lond. Math. Soc. 11 (1974), 509–512.Google Scholar
18.Mason, A. W., A further note on subgroups of GL(n, A) which are generated by commutators, Arch. Math. 37(5) (1981), 401–405.CrossRefGoogle Scholar
19.Mason, A. W., On subgroups of GL(n, A) which are generated by commutators, II, J. Reine Angew. Math. 322 (1981), 118–135.Google Scholar
20.Mason, A. W. and Stothers, W. W., On subgroup of GL(n, A) which are generated by commutators, Invent. Math. 23 (1974), 327–346.CrossRefGoogle Scholar
21.Nica, B., A true relative of Suslin's normality theorem, Enseign. Math. 61(1–2) (2015), 151–159.CrossRefGoogle Scholar
22.Stein, M. R., Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93(4) (1971), 965–1004.CrossRefGoogle Scholar
23.Steinberg, R., Lectures on Chevalley groups (Yale University, 1967).Google Scholar
24.Stepanov, A., Elementary calculus in Chevalley groups over rings, J. Prime Res. Math. 9 (2013), 79–95.Google Scholar
25.Stepanov, A. V., Non-abelian K-theory for Chevalley groups over rings, J. Math. Sci. 209(4) (2015), 645–656.CrossRefGoogle Scholar
26.Stepanov, A., Structure of Chevalley groups over rings via universal localization, J. Algebra 450 (2016), 522–548.CrossRefGoogle Scholar
27.Stepanov, A. and Vavilov, N., Decomposition of transvections: a theme with variations, K-Theory 19(2) (2000), 109–153.CrossRefGoogle Scholar
28.Stepanov, A. and Vavilov, N., On the length of commutators in Chevalley groups, Israel J. Math. 185 (2011), 253–276.CrossRefGoogle Scholar
29.Suslin, A. A., The structure of the special linear group over polynomial rings, Math. USSR Izv. 11(2) (1977), 235–253.CrossRefGoogle Scholar
30.Taddei, G., Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Contemp. Math. 55(II) (1986), 693–710.CrossRefGoogle Scholar
31.Tits, J., Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér. A 283 (1976), 693–695.Google Scholar
32.Vaserstein, L. N., On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 36(5) (1986), 219–230.CrossRefGoogle Scholar
33.Vavilov, N., Structure of Chevalley groups over commutative rings, in Proceedings of the Conference on Non-associative Algebras and Related Topics (Hiroshima, 1990), pp. 219–335 (World Scientific Publishing, London, 1991).Google Scholar
34.Vavilov, N., Unrelativised standard commutator formula, J. Math. Sci. New York 24(4) (2019), 527–534.CrossRefGoogle Scholar
35.Vavilov, N. and Plotkin, E., Chevalley groups over commutative rings I: elementary calculations, Acta Appl. Math. 45 (1996), 73–113.CrossRefGoogle Scholar
36.Vavilov, N. A. and Stepanov, A. V., Standard commutator formula, Vestnik St. Petersburg State Univ. Ser. 1 41(1) (2008), 5–8.Google Scholar
37.Vavilov, N. A. and Stepanov, A. V., Standard commutator formula, revisited, Vestnik St. Petersburg State Univ. Ser. 1 43(1) (2010), 12–17.Google Scholar
38.You, H., On subgroups of Chevalley groups which are generated by commutators, J. Northeast Norm. Univ. 2 (1992), 9–13.Google Scholar