Simple algebraic and semialgebraic groups over real closed fields

Peterzil, Ya’acov; Pillay, Anand; Starchenko, Sergei

doi:10.1090/S0002-9947-00-02667-2

Simple algebraic and semialgebraic groups over real closed fields
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by Ya’acov Peterzil, Anand Pillay and Sergei Starchenko PDF

Trans. Amer. Math. Soc. 352 (2000), 4421-4450 Request permission

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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