A definability criterion for connected Lie groups

Abstract

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239-255)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 (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441-452) 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 its Levi subgroups with finite center, 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 (Conversano, Onshuus,