We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R = ℝ, this result was claimed in [5], but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.
Ehud Hrushovski 1 ; Anand Pillay 1
@article{CML_2011__3_4_577_0,
author = {Ehud Hrushovski and Anand Pillay},
title = {Affine {Nash} groups over real closed fields},
journal = {Confluentes Mathematici},
pages = {577--585},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {4},
year = {2011},
doi = {10.1142/S179374421100045X},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S179374421100045X/}
}
TY - JOUR AU - Ehud Hrushovski AU - Anand Pillay TI - Affine Nash groups over real closed fields JO - Confluentes Mathematici PY - 2011 SP - 577 EP - 585 VL - 3 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S179374421100045X/ DO - 10.1142/S179374421100045X LA - en ID - CML_2011__3_4_577_0 ER -
%0 Journal Article %A Ehud Hrushovski %A Anand Pillay %T Affine Nash groups over real closed fields %J Confluentes Mathematici %D 2011 %P 577-585 %V 3 %N 4 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S179374421100045X/ %R 10.1142/S179374421100045X %G en %F CML_2011__3_4_577_0
Ehud Hrushovski; Anand Pillay. Affine Nash groups over real closed fields. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 577-585. doi : 10.1142/S179374421100045X. https://cml.centre-mersenne.org/articles/10.1142/S179374421100045X/
[1] M. Artin and B. Mazur, On periodic points, Ann. Math. 81 (1965) 82–99.
[2] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry (Springer, 1998).
[3] A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, preprint 2011. http://www.maths.leeds.ac.uk/∼pillay/
[4] L. van den Dries, Tame Topology and o-Minimal Structures, LMS Lecture Note Series, Vol. 248 (Cambridge Univ. Press, 1998).
[5] E. Hrushovski and A. Pillay, Groups definable in local fields and pseudofinite fields, Israel J. Math. 85 (1994) 203–262.
[6] A. Pillay, On groups and fields definable in o-minimal structures, J. Pure Appl. Alg. 53 (1988) 239–255.
[7] M. Shiota, Nash Manifolds, Lecture Notes in Mathematics, Vol. 1269 (Springer, 1987).
[8] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Prentice-Hall, 1974).
Cité par Sources :