Definability in the group of infinitesimals of a compact Lie group

Bays, Martin; Peterzil, Ya’acov

doi:10.5802/cml.58

Bays, Martin ¹ ; Peterzil, Ya’acov ²

¹ Institut für Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
² Department of Mathematics, University of Haifa, Haifa, Israel

Confluentes Mathematici, Tome 11 (2019) no. 2, pp. 3-23.

Résumé

We show that for $G$ a simple compact Lie group, the infinitesimal subgroup $G^{00}$ is bi-interpretable with a real closed convexly valued field. We deduce that for $G$ an infinite definably compact group definable in an o-minimal expansion of a field, $G^{00}$ is bi-interpretable with the disjoint union of a (possibly trivial) $ℚ$ -vector space and finitely many (possibly zero) real closed valued fields. We also describe the isomorphisms between such infinitesimal subgroups, and along the way prove that every definable field in a real closed convexly valued field $R$ is definably isomorphic to $R$ .

Reçu le : 2019-02-04
Révisé le : 2019-07-22
Accepté le : 2019-08-06
Publié le : 2020-03-09

DOI : 10.5802/cml.58

Classification : 03C64, 22E15
Mots clés : Model Theory, Compact Lie Group, Infinitesimal Subgroup, O-Minimality, Bi-interpretation, Valued Field

Affiliations des auteurs :

Bays, Martin ¹ ; Peterzil, Ya’acov ²

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Bays, Martin; Peterzil, Ya’acov. Definability in the group of infinitesimals of a compact Lie group. Confluentes Mathematici, Tome 11 (2019) no. 2, pp. 3-23. doi : 10.5802/cml.58. http://www.numdam.org/articles/10.5802/cml.58/

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