A definability criterion for connected Lie groups

Abstract

It has been known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in \cite{COSsolvable} in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group $G$ is Lie isomorphic to a group definable in an $o$-minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately we cannot achieve a full characterization. Assuming that a Lie group $G$ has a "good Levi descomposition", we prove that in order for $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in \cite{COSsolvable}.