A definability criterion for connected Lie groups

TL;DR

This paper investigates when connected Lie groups can be realized as groups definable in o-minimal structures, providing a complete classification for linear groups and partial results for general cases based on Levi decompositions.

Contribution

It offers a complete classification for linear Lie groups and establishes criteria for general connected Lie groups with good Levi decompositions to be definable.

Findings

Linear Lie groups are definable iff their solvable radical is definable.

Connected Lie groups with good Levi decompositions are definable iff their solvable radical is definable.

Partial characterization for general connected Lie groups based on Levi decompositions.

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}.