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Simple algebraic and semialgebraic groups over real closed fields
Author(s):
Ya'acov
Peterzil;
Anand
Pillay;
Sergei
Starchenko
Journal:
Trans. Amer. Math. Soc.
352
(2000),
4421-4450.
MSC (1991):
Primary 03C64, 22E15, 20G20;
Secondary 12J15
Posted:
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
Email:
kobi@mathcs2.haifa.ac.il
Anand
Pillay
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801
Email:
pillay@math.uiuc.edu
Sergei
Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, CCMB, Notre Dame, Indiana 46556
Email:
starchenko.1@nd.edu
DOI:
10.1090/S0002-9947-00-02667-2
PII:
S 0002-9947(00)02667-2
Received by editor(s):
February 25, 1998
Posted:
June 13, 2000
Additional Notes:
We thank the referee for valuable comments
The second and the third authors were partially supported by NSF
Copyright of article:
Copyright
2000,
American Mathematical Society
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