Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. O. Belegradek, On mutual interpretation of a ring $R$and the group $UT_n(R)$, preprint.
  • 2. A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Math., 126 (Spinger-Verlag, 1991). MR 92d:20001
  • 3. A. Borel and J. Tits, Homomorphismes ``abstraits'' de groupes algébraic simples, Annals of Math. 97 (1973), 499-571. MR 47:5134
  • 4. A. Borovik and A. Nesin, Groups of finite Morley rank, (The Clarendon Press, Oxford University Press, New York, 1994). MR 96c:20004
  • 5. L. van den Dries, O-minimal structures, Logic: From foundations to applications, (ed: Hodges, Steinhorn, Truss), Oxford University Press, 1996, 137-185. MR 98b:03053
  • 6. L. van den Dries Tame topology and o-minimal structures, Cambridge University Press, Cambridge, 1998. MR 99j:03001
  • 7. H. Freudenthal, Die Topologie der Lieschen gruppen als algebraisches Phänomenä. I, Ann. of Math 42 (1941), 1051-1074. Erratum ibid. 47 (1946), 829-830. MR 3:198a
  • 8. W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42 (Cambridge University Press, Cambridge, 1993). MR 94e:03002
  • 9. E. Hrushovski and A. Pillay, Groups definable in local fields and pseudo-finite fields, Israel Journal of Mathematics 85 (1994), 203-262. MR 95f:12015
  • 10. D. Marker, Semialgebraic expansions of $\mathbb{C} $, Trans. Amer. Math. Soc. 320 (1990), 581-592. MR 90k:03034
  • 11. A. Nesin and A. Pillay, Some model theory of compact Lie groups, Trans. Amer. Math. Soc. 326 (1991), 453-463. MR 91j:03038
  • 12. M. Otero, Y. Peterzil and A. Pillay, Groups and rings definable in o-minimal expansions of real closed fields, Bull. London Math. Soc. 28 (1993) 7-14. MR 96i:12006
  • 13. Y. Peterzil, Constructing a group interval in o-minimal structures, J. of Pure and Applied Algebra 94 (1994), 85-100. MR 95h:03085
  • 14. Y. Peterzil, A. Pillay and S. Starchenko, Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352 (2000), 4397-4419. CMP 99:17
  • 15. Y. Peterzil and S. Starchenko, A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society (3) 77 (1998), 481-523. MR 2000b:03123
  • 16. A. Pillay, On groups definable in o-minimal structures, J. of Pure and Applied Algebra 53 (1988), 239-255. MR 59i:03069
  • 17. A. Pillay, An application of model theory to real and p-adic algebraic groups, J. of Algebra 126 (1989), 139-146. MR 90m:03061
  • 18. V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, v. 139., Academic Press, Inc. MR 95b:11039
  • 19. B. Poizat, Group Stables, (Nur al-Mantiq wal-Ma'rifah, 1987). MR 89b:03056
  • 20. A. Strzebonski, Euler characteristic in semialgebraic and other o-minimal groups, J. Pure Appl. Algebra 96 (1994), 173-201. MR 95j:03067
  • 21. J. Tits, Homomorphismes et automorphismes ``abstraits'' de groupes algébriques et arithmétique, Actes du Congrès Internat. des Math., Tome 2, 1970, 349-355. MR 55:5760
  • 22. B. Weisfeiler, On abstract homomorphisms of anisotropic algebraic groups over real-closed fields, Journal of Algebra 60 (1979), 485-519. MR 80k:20045

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03C64, 22E15, 20G20, 12J15

Retrieve articles in all journals with MSC (1991): 03C64, 22E15, 20G20, 12J15

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

American Mathematical Society