Abstract:
Let GG be a not necessarily split reductive group scheme over a commutative ring RR with 11. Given a parabolic subgroup PP of GG, the elementary group EP(R)EP(R) is defined to be the subgroup of G(R)G(R) generated by UP(R)UP(R) and UP−(R)UP−(R), where UPUP and UP−UP− are the unipotent radicals of PP and its opposite P−P− respectively. It is proved that if GG contains a Zariski locally split torus of rank 2, then the group EP(R)=E(R)EP(R)=E(R) does not depend on PP, and, in particular, is normal in G(R)G(R).
Keywords:
Reductive group scheme, elementary subgroup, Whitehead group, parabolic subgroup.
Citation:
V. Petrov, A. Stavrova, “Elementary subgroups of isotropic reductive groups”, Algebra i Analiz, 20:4 (2008), 160–188; St. Petersburg Math. J., 20:4 (2009), 625–644
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\paper Elementary subgroups of isotropic reductive groups
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 4
\pages 160--188
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\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 4
\pages 625--644
\crossref{https://doi.org/10.1090/S1061-0022-09-01064-4}
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Linking options:
https://www.mathnet.ru/eng/aa525
https://www.mathnet.ru/eng/aa/v20/i4/p160
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E. B. Plotkin, A. I. Generalov, N. S. Geldkhauzer, N. L. Gordeev, A. Yu. Luzgarev, V. V. Nesterov, I. A. Panin, V. A. Petrov, S. Yu. Pilyugin, A. V. Stepanov, A. K. Stavrova, V. G. Khalin, “O Nikolae Aleksandroviche Vavilove”, Voprosy teorii predstavlenii algebr i grupp. 40, Posvyaschaetsya pamyati Nikolaya Aleksandrovicha VAVILOVA, Zap. nauchn. sem. POMI, 531, POMI, SPb., 2024, 7–40
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E. Yu. Voronetskii, “Twisted forms of classical groups”, St. Petersburg Math. J., 34:2 (2023), 179–204
Voronetsky E., “Centrality of K-2-Functor Revisited”, J. Pure Appl. Algebr., 225:4 (2021), 106547
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Stavrova A., “Chevalley Groups of Polynomial Rings Over Dedekind Domains”, J. Group Theory, 23:1 (2020), 121–132
Ambily A.A., Rao R.A., “Normality of Dser Elementary Orthogonal Group”, J. Pure Appl. Algebr., 224:7 (2020), UNSP 106283
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Boelaert L., De Medts T., Stavrova A., “Moufang Sets and Structurable Division Algebras”, Mem. Am. Math. Soc., 259:1245 (2019), 1+
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