Cartan subgroups and regular points of o‐minimal groups

Abstract

Let G be a group definable in an o‐minimal structure scriptM. We prove that the union of the Cartan subgroups of G is a dense subset of G. When scriptM is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.

The derived subgroup of linear and simply-connected o-minimal groups

“…Suppose first that G ≤ GL(n, R) is linear. Let g = r + s be a Levi decomposition of the Lie algebra g of G, where r denotes the radical of g. We note that Lie(R(G)) = r. Indeed, since R(G) is solvable, its Lie algebra Lie(R(G)) is solvable [3,Lem.3.7] and therefore Lie(R(G)) ⊆ r.…”

“…A Lie subalgebra of gl(n, R) is said to be algebraic if it is the Lie algebra of an algebraic subgroup of GL(n, R). Given a Lie subalgebra g of gl(n, R), a(g) denotes the minimal algebraic Lie subalgebra of gl(n, R) containing g. We recall that if g is a subalgebra of gl(n, R) then [g, g] = [a(g), a(g)] is algebraic (see [17,Ch.3,§3]).…”

“…For, in that case the quotient Suppose first that G ≤ GL(n, R) is linear. Let g = r + s be a Levi decomposition of the Lie algebra g of G, where r denotes the radical of g. We note that Lie(R(G)) = r. Indeed, since R(G) is solvable, its Lie algebra Lie(R(G)) is solvable [3,Lem.3.7] and therefore Lie(R(G)) ⊆ r. In particular, since G/R(G) is semisimple, it follows from Fact 2.6 that g/Lie(R(G)) is semisimple and so Lie(R(G)) = r. On the other hand, since s = [s, s] = [a(s), a(s)] is algebraic, there is an algebraic subgroup S 1 of GL(n, R) whose Lie algebra is s. Therefore S := S o 1 is a connected semisimple definable subgroup of G such that G = R(G)S and R(G) ∩ S is finite, as desired. Now, suppose that G is almost-linear, i.e.…”

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.…”