2023/02/08 by Onshuus, Alf
#03C64 #22E15 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2302.04251
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 (\citePi88) that any definable group is a Lie group, and in \citeCOP a complete characterization of when a Lie group is 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 (\citeWilkie) 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 G1, G2, 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 RPfaff that makes it an analytic group.