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Elementary subgroup of an isotropic reductive group is perfect


Authors: A. Yu. Luzgarev and A. K. Stavrova
Translated by: A. Yu. Luzgarev
Original publication: Algebra i Analiz, tom 23 (2011), nomer 5.
Journal: St. Petersburg Math. J. 23 (2012), 881-890
MSC (2010): Primary 20G35
Published electronically: July 10, 2012
MathSciNet review: 2918426
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Abstract: Let $ G$ be an isotropic reductive algebraic group over a commutative ring $ R$. Assume that the elementary subgroup $ E(R)$ of the group of points $ G(R)$ is well defined. Then $ E(R)$ is perfect, except for the well-known case of a split reductive group of type $ C_2$ or $ G_2$.


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Additional Information

A. Yu. Luzgarev
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: mahalex@gmail.com

A. K. Stavrova
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Germany
Email: anastasia.stavrova@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2012-01221-5
Keywords: Reductive group, affine group scheme, elementary subgroup, Chevalley commutator formula
Received by editor(s): May 27, 2010
Published electronically: July 10, 2012
Additional Notes: Supported by the research program 6.38.74.2011 “The Structural Theory and Geometry of Algebraic Groups and Their Applications in Representation Theory and Algebraic K-Theory” of St. Petersburg State University and by the RFBR projects 09-01-00878, 09-01-90304, 10-01-00551, 10-01-90016. The first author was also supported by the RFBR project 09-01-00874. The second author was also supported by the grants DFG GI 706/1-2 and DFG SFB/TR 45.
Dedicated: To our Teacher Nikolai Vavilov, on the occasion of his 60th birthday
Article copyright: © Copyright 2012 American Mathematical Society