It has been known for some time that any group definable in an o-minimal expansion of the real field can be endowed definably with the structure of a Lie group, and that any definable homomorphisms between definable groups is a Lie homomorphism (under the above mentioned Lie structure). In this talk we explore the converse: We will characterize when a Lie group has a Lie isomorphic group which is definable in an o-minimal expansion of the real field, when Lie isomorphisms between such definable groups is definable, and whether one can achieve a definable Lie analytic structure in any such definable group.
