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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2302.04251 |
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| _version_ | 1866909510946258944 |
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| author | Onshuus, Alf |
| author_facet | Onshuus, Alf |
| contents | 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''. With this terminology, it is known (\cite{Pi88}) that any definable group is a Lie group, and in \cite{COP} a complete characterization of when a Lie group is \emph{Lie isomorphic} to a definable group'' was given. We continue the analysis by explaining when a Lie isomorphism between definable groups is definable.
Among other things, we generalize Wilkie's result on the o-minimality of the exponential function (\cite{Wilkie}) by completely characterizing when, given an o-minimal expansion $\mathcal R$ of the real field and a Lie isomorphisms $ϕ$ between two $\mathcal R$-definable groups $G_1, G_2$, $ϕ$ can be added to the language of $\mathcal R$ preserving o-minimality. We also prove 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04251 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuous isomorphisms between groups definable in o-minimal expansions of the real field Onshuus, Alf Logic 03C64, 22E15 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''. With this terminology, it is known (\cite{Pi88}) that any definable group is a Lie group, and in \cite{COP} a complete characterization of when a Lie group is \emph{Lie isomorphic} to a definable group'' was given. We continue the analysis by explaining when a Lie isomorphism between definable groups is definable. Among other things, we generalize Wilkie's result on the o-minimality of the exponential function (\cite{Wilkie}) by completely characterizing when, given an o-minimal expansion $\mathcal R$ of the real field and a Lie isomorphisms $ϕ$ between two $\mathcal R$-definable groups $G_1, G_2$, $ϕ$ can be added to the language of $\mathcal R$ preserving o-minimality. We also prove 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. |
| title | Continuous isomorphisms between groups definable in o-minimal expansions of the real field |
| topic | Logic 03C64, 22E15 |
| url | https://arxiv.org/abs/2302.04251 |