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Cartan subgroups of groups definable in o-minimal structures

Part of: Topological and differentiable algebraic systems Model theory

Published online by Cambridge University Press:  28 November 2013

Elías Baro ,
Eric Jaligot  and
Margarita Otero
Elías Baro
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Eric Jaligot
Affiliation:
Institut Fourier, CNRS, Université Grenoble I, 100 rue des maths, BP 74, 38402 St Martin d’Hères cedex, France
Margarita Otero
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain ([email protected])
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Abstract

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.

Keywords

Lie groupssemialgebraic groupsgroups definable in o-minimal structuresCartan subgroups

MSC classification

Secondary: 22A05: Structure of general topological groups 03C64: Model theory of ordered structures; o-minimality
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 849 - 893
Copyright
©Cambridge University Press 2013 

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References

Baro, E., Jaligot, E. and Otero, M., Commutators in groups definable in o-minimal structures, Proc. Amer. Math. Soc. 140 (10) (2012), 3629–3643.CrossRefGoogle Scholar
Berarducci, A., Zero-groups and maximal tori, in Logic colloquium 2004, Lect. Notes Log., Volume 29, pp. 33–46 (Assoc. Symbol. Logic, Chicago, IL, 2008).Google Scholar
Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer, New York, 1991).CrossRefGoogle Scholar
Borovik, A. and Nesin, A., Groups of finite Morley rank, Oxford Logic Guides, Volume 26 (The Clarendon Press Oxford University Press, New York, 1994), Oxford Science Publications.CrossRefGoogle Scholar
Bourbaki, N., Lie groups and Lie algebras, Elements of Mathematics (Berlin) (Springer, Berlin, 1998), Chapters 1–3 (transl. from French), Reprint of the 1989 English translation.Google Scholar
Bryant, R. M., Groups with the minimal condition on centralizers, J. Algebra 60 (2) (1979), 371–383.CrossRefGoogle Scholar
Carter, R. W., Nilpotent self-normalizing subgroups of soluble groups, Math. Z. 75 (1960/1961), 136–139.CrossRefGoogle Scholar
Chevalley, C., Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Actualités Sci. Ind., Volume 1226 (Hermann & Cie, Paris, 1955).Google Scholar
Deloro, A. and Jaligot, E., Groups of finite Morley rank with solvable local subgroups, Commun. Algebra 40 (3) (2012), 1019–1068.CrossRefGoogle Scholar
Edmundo, M. J., Solvable groups definable in o-minimal structures, J. Pure Appl. Algebra 185 (1–3) (2003), 103–145.CrossRefGoogle Scholar
Edmundo, M. J., A remark on divisibility of definable groups, MLQ Math. Log. Q. 51 (6) (2005), 639–641.CrossRefGoogle Scholar
Eleftheriou, P., Peterzil, Y. and Ramakrishnan, J., Interpretable groups are definable, preprint, 2013.CrossRefGoogle Scholar
Frécon, O., Sous-groupes anormaux dans les groupes de rang de Morley fini résolubles, J. Algebra 229 (1) (2000), 118–152.CrossRefGoogle Scholar
Frécon, O., Around unipotence in groups of finite Morley rank, J. Group Theory 9 (3) (2006), 341–359.CrossRefGoogle Scholar
Frécon, O. and Jaligot, E., Conjugacy in groups of finite Morley rank, in Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., Volume 350, pp. 1–58 (Cambridge University Press, Cambridge, 2008).Google Scholar
Harish-Chandra, , The characters of semisimple Lie groups, Trans. Amer. Math. Soc. 83 (1956), 98–163.CrossRefGoogle Scholar
Harish-Chandra, , Some results on an invariant integral on a semisimple Lie algebra, Ann. of Math. (2) 80 (1964), 551–593.CrossRefGoogle Scholar
Hofmann, K. H., Near-Cartan algebras and groups, Sem. Sophus Lie 2 (2) (1992), 135–151.Google Scholar
Hrushovski, E., Peterzil, Y. and Pillay, A., On central extensions and definably compact groups in o-minimal structures, J. Algebra 327 (2011), 71–106.CrossRefGoogle Scholar
Jaligot, E., Generix never gives up, J. Symbolic Logic 71 (2) (2006), 599–610.CrossRefGoogle Scholar
Jaligot, E., Cosets, genericity, and the Weyl group, J. Algebra 322 (4) (2009), 1060–1071.CrossRefGoogle Scholar
Luke, G. L.  (ed.) Representation theory of Lie groups. London Mathematical Society Lecture Note Series, Volume 34 (Cambridge University Press, Cambridge, 1979).Google Scholar
Neeb, K.-H., Weakly exponential Lie groups, J. Algebra 179 (2) (1996), 331–361.CrossRefGoogle Scholar
Otero, M., A survey on groups definable in o-minimal structures, in Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., Volume 350, pp. 177–206 (Cambridge University Press, Cambridge, 2008).Google Scholar
Peterzil, Y., Pillay’s conjecture and its solution—a survey, in Logic colloquium 2007, Lect. Notes Log., Volume 35, pp. 177–203 (Assoc. Symbol. Logic, La Jolla, CA, 2010).CrossRefGoogle Scholar
Peterzil, Y., Pillay, A. and Starchenko, S., Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352 (10) (2000), 4397–4419.CrossRefGoogle Scholar
Peterzil, Y., Pillay, A. and Starchenko, S., Simple algebraic and semialgebraic groups over real closed fields, Trans. Amer. Math. Soc. 352 (10) (2000), 4421–4450, electronic.CrossRefGoogle Scholar
Peterzil, Y., Pillay, A. and Starchenko, S., Linear groups definable in o-minimal structures, J. Algebra 247 (1) (2002), 1–23.CrossRefGoogle Scholar
Peterzil, Y. and Starchenko, S., Definable homomorphisms of abelian groups in o-minimal structures, Ann. Pure Appl. Logic 101 (1) (2000), 1–27.CrossRefGoogle Scholar
Pillay, A., On groups and fields definable in o-minimal structures, J. Pure Appl. Algebra 53 (3) (1988), 239–255.CrossRefGoogle Scholar
Poizat, B., Stable groups, Mathematical Surveys and Monographs, Volume 87 (American Mathematical Society, Providence, RI, 2001), Translated from the 1987 French original by Moses Gabriel Klein.CrossRefGoogle Scholar
van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, Volume 248 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Varadarajan, V. S., Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis (Prentice-Hall Inc, Englewood Cliffs, N.J, 1974).Google Scholar
Warner, G., Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Volume Band 188 (Springer, New York, 1972).Google Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (4) (1996), 1051–1094.CrossRefGoogle Scholar
Winkelmann, J.., Generic subgroups of Lie groups, Topology 41 (1) (2002), 163–181.CrossRefGoogle Scholar