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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Simple algebraic and semialgebraic groups over real closed fields

Authors: Ya'acov Peterzil, Anand Pillay and Sergei Starchenko
Journal: Trans. Amer. Math. Soc. 352 (2000), 4421-4450
MSC (1991): Primary 03C64, 22E15, 20G20; Secondary 12J15
Published electronically: June 13, 2000
MathSciNet review: 1779482
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Abstract: We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

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Additional Information

Ya'acov Peterzil
Affiliation: Department of Mathematics and Computer Science, Haifa University, Haifa, Israel

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801

Sergei Starchenko
Affiliation: Department of Mathematics, University of Notre Dame, CCMB, Notre Dame, Indiana 46556

Received by editor(s): February 25, 1998
Published electronically: June 13, 2000
Additional Notes: We thank the referee for valuable comments
The second and the third authors were partially supported by NSF
Article copyright: © Copyright 2000 American Mathematical Society

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