Definability of continuous isomorphisms of groups definable in o-minimal expansions of the real field

• Published in: arXiv.org
• Main Author: Onshuus, Alf
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 17.07.2024
• Subjects: Homomorphisms; Isomorphism; Lie groups
• ISSN: 2331-8422

In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer to as "definable groups"). It is known (\cite{Pi88}) that any group definable in an o-minimal expansion of the real field is a Lie group, and in \cite{COP} a complete characterization of when a Lie group has a "definable group" which is \emph{Lie isomorphic} to it was given. We continue the analysis by explaining when a Lie homomorphism between definable groups is a definable isomorphism. Among other things, we prove that in any o-minimal expansion \(\mathcal R\) of the real field we can add a function symbol for any Lie isomorphism between definable groups to the language of \(\mathcal R\) preserving o-minimality, and that any definable group \(G\) can be endowed with an analytic manifold structure definable in \(\mathcal R_{\text{Pfaff}}\) that makes it an analytic group.