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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Elementary subgroups of isotropic reductive groups

Authors: V. Petrov and A. Stavrova
Translated by: The authors
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 625-644
MSC (2000): Primary 20G35
Published electronically: June 2, 2009
MathSciNet review: 2473747
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Abstract: Let $ G$ be a not necessarily split reductive group scheme over a commutative ring $ R$ with $ 1$. Given a parabolic subgroup $ P$ of $ G$, the elementary group $ {E}_P(R)$ is defined to be the subgroup of $ G(R)$ generated by $ {U}_P(R)$ and $ {U}_{P^-}(R)$, where $ {U}_P$ and $ {U}_{P^-}$ are the unipotent radicals of $ P$ and its opposite $ P^-$, respectively. It is proved that if $ G$ contains a Zariski locally split torus of rank 2, then the group $ {E}_P(R) ={E}(R)$ does not depend on $ P$, and, in particular, is normal in $ G(R)$.

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Additional Information

V. Petrov
Affiliation: University of Alberta, Edmonton, Canada

A. Stavrova
Affiliation: St. Petersburg State University, St. Petersburg, Russia

Keywords: Reductive group scheme, elementary subgroup, Whitehead group, parabolic subgroup
Received by editor(s): December 21, 2007
Published electronically: June 2, 2009
Additional Notes: The first author was supported by PIMS Postdoctoral Fellowship and by INTAS (grant no. 03-51-3251)
Article copyright: © Copyright 2009 American Mathematical Society

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