Lie groups and definability

Abstract

It is known since 1988 (Pillay) that any group definable in an o-minimal expansion of the real field is a Lie group. In this work we give criteria for Lie groups to have a definable copy, that is to be isomorphic (as a Lie group) to a group definable in such expansions. More particularly, we show that under the criterions given by Conversano, Onshuus and Starchenko for the solvable case, the group is actually isomorphic to a matrix group definable using only the real exponential and the field structure. Morover we characterize completely those linear Lie grousp with definable copies. This characterization extends to the nonb linear case if the Lie group has Levi subgroup with finite center.