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SOLVABLE LIE GROUPS DEFINABLE IN O-MINIMAL THEORIES

Part of: Model theory Ordered structures Lie groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  28 April 2016

Annalisa Conversano ,
Alf Onshuus  and
Sergei Starchenko
Annalisa Conversano
Affiliation:
Massey University Albany, INMS, IIMS Building, Private Bag 102904, North Shore City 0745, New Zealand ([email protected])
Alf Onshuus
Affiliation:
Departamento de Matemáticas, Universidad de los Andes, Cra 1 No. 18A-10, Edificio H, Bogotá 111711, Colombia ([email protected])
Sergei Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA ([email protected])
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Abstract

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

Keywords

definable groupscompletely solvable torsion-free groupso-minimal group cohomologyLie groupsLie algebras

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality 20J06: Cohomology of groups 06F25: Ordered rings, algebras, modules 22E25: Nilpotent and solvable Lie groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 2 , April 2018 , pp. 441 - 452
Copyright
© Cambridge University Press 2016 

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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. MR 2929031.Google Scholar
Borel, A., Linear Algebraic Groups, Second, Graduate Texts in Mathematics, 126 (Springer, New York, 1991). MR 1102012 (92d:20001).Google Scholar
Conversano, A. and Pillay, A., Connected components of definable groups and o-minimality I, Adv. Math. 231(2) (2012), 605–623. MR 2955185.CrossRefGoogle Scholar
Dixmier, J., L’application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. France 85 (1957), 113–121. MR 0092930 (19,1182a).Google Scholar
Edmundo, M. J., Solvable groups definable in o-minimal structures, J. Pure Appl. Algebra 185(1–3) (2003), 103–145. MR 2006422 (2004j:03048).Google Scholar
Gorbatsevich, V. V., Onishchik, A. L. and Vinberg, È. B., Structure of Lie groups and Lie algebras, a translation of ıt current problems in mathematics. Fundamental directions, in Lie Groups and Lie Algebras, III (ed. Vinberg, È. B.), Encyclopaedia of Mathematical Sciences, Volume 41, p. iv+248 (Springer-Verlag, Berlin, 1994).Google Scholar
Iwasawa, K., On some types of topological groups, Ann. of Math. 50(3) (1949), 507–558.CrossRefGoogle Scholar
Knapp, A. W., Lie Groups Beyond an Introduction, Second, Progress in Mathematics, 140 (Birkhäuser Boston, Inc., Boston, MA, 2002). MR 1920389 (2003c:22001).Google Scholar
Miller, C. and Starchenko, S., A growth dichotomy for o-minimal expansions of ordered groups, Trans. Amer. Math. Soc. 350(9) (1998), 3505–3521. MR 1491870 (99e:03025).CrossRefGoogle Scholar
Onishchik, A. L. and Vinberg, È. B., Foundations of Lie Theory, Lie Groups and Lie Algebras, I, pp. 1–94. (1993).Google Scholar
Peterzil, Y., Pillay, A. and Starchenko, S., Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352(10) (2000), 4397–4419. MR 1707202 (2001b:03036).Google Scholar
Peterzil, Y., Pillay, A. and Starchenko, S., Linear groups definable in o-minimal structures, J. Algebra 247(1) (2002), 1–23. MR 1873380 (2002i:03043).CrossRefGoogle Scholar
Peterzil, Y. and Starchenko, S., On torsion-free groups in o-minimal structures, Illinois J. Math. 49(4) (2005), 1299–1321. (electronic). MR 2210364 (2007b:03058).Google Scholar
Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, J. Lond. Math. Soc. (2) 59(3) (1999), 769–786. MR 1709079 (2000i:03055).Google Scholar
Pillay, A., On groups and fields definable in o-minimal structures, J. Pure Appl. Algebra 53(3) (1988), 239–255. MR 961362 (89i:03069).Google Scholar
Strzebonski, A. W., One-dimensional groups definable in o-minimal structures, J. Pure Appl. Algebra 96(2) (1994), 203–214. MR 1303546 (95j:03068).Google Scholar
van den Dries, L., Tame Topology and o-minimal Structures, London Mathematical Society Lecture Note Series, 248 (Cambridge University Press, Cambridge, 1998). MR 1633348 (99j:03001).Google Scholar
van den Dries, L., Macintyre, A. and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140(1) (1994), 183–205. MR 1289495 (95k:12015).Google Scholar
van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Math. J. 84(2) (1996), 497–540.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. MR 1398816 (98j:03052).CrossRefGoogle Scholar