Weak quadratic overgroups for type I solvable lie groups of the form ℝ ⋉ ℝn

Abstract

Let G be a type I connected and simply connected solvable Lie group defined as the semi-direct product of ℝ and an n-dimensional Abelian ideal N for some n ≥ 1. Let g*/G denote the set of coadjoint orbits of G, where g* is the dual vector space of the Lie algebra g of G. Generally, the closed convex hull of a coadjoint orbit O ⊂ g* does not characterize O. However, we say that a subset X in g*/G is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in X. In this paper, ourmain result provides an explicit construction of an overgroup, denoted G⁺, containing G as a subgroup and a quadratic map ϕ sending each G-orbit in g* to G⁺-orbit in (g⁺)*, in such a manner that the set ϕ(g*)/G⁺ is convex hull separable, which leads to the separation of elements of g*/G. The Lie group G⁺ is called a weak quadratic overgroup for G.