A definability criterion for connected Lie groups

Abstract

It is known since [7] that any group definable in an o-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to ask when does a Lie group is Lie isomorphic to a group definable in such expansion. Conversano, Starchenko and the first author answered this question in [2] in the case where the group is solvable. We give here a criterion in the case where the group is linear. More precisely if G is a linear Lie group it is isomorphic to a group definable in an o-minimal expansion of the reals if and only if its solvable radical is isomorphic to such group.

Real Lie groups and o-minimality

“…Proof. (⇒) If S does not have a definable copy, then Z(S) is infinite, by [9,Theo 3]. Therefore S/(R ∩ S) has infinite center too, since R ∩ S is finite.…”

“…It follows that S/(R ∩ S) ∼ = G/R does not have a definable copy, contradiction. (⇐) If S has a definable copy, then Z(S) is finite, and G has a definable copy again by [9,Theo 3].…”

“…In Theorem 4.1 we complete the research started in [9] to prove that the necessary conditions identified in the work mentioned above are sufficient. Namely, that if a Lie group G satisfies:…”

Groups definable in o-minimal structures: various properties and a diagram

“…Further work on definable groups outside the scope of this diagram can be found in [4,7,8,18,22,35,36,39,34,40,43,44,46,47,48,50,51,69,73]. In recent years, the investigation has been extended by several authors to the wider class of locally definable groups.…”