Статьи, опубликованные в английской версии журнала
Weak Quadratic Overgroups for Type I Solvable Lie Groups of the Form $\mathbb{R}\ltimes\mathbb{R}^{n}$
L. Abdelmoula,
Y. Bouaziz Sfax University, Sfax, Tunisia
Аннотация:
Let
$G$ be a type I connected and simply connected solvable Lie group defined as the semi-direct product of
$\mathbb{R}$ and an
$n$-dimensional Abelian ideal
$N$ for some
$n\geq 1.$ Let
$\mathfrak{g}^*/G$ denote the set of coadjoint orbits of
$G$, where
$\mathfrak{g}^*$ is the dual vector space of the Lie algebra
$\mathfrak{g}$ of
$G.$ Generally, the closed convex hull of a coadjoint orbit
$\mathcal{O}\subset \mathfrak{g}^*$ does not characterize
$\mathcal{O}.$ However, we say that a subset
$X$ in
$\mathfrak{g}^*/G$ is
convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in
$X.$ In this paper, our main result provides an explicit construction of an overgroup, denoted
$G^+,$ containing
$G$ as a subgroup and a quadratic map
$\varphi$ sending each
$G$-orbit in
$\mathfrak{g}^*$ to
$G^+$-orbit in
$(\mathfrak{g}^+)^*,$ in such a manner that the set
$\varphi(\mathfrak{g}^*)/G^+$ is convex hull separable, which leads to the separation of elements of
$\mathfrak{g}^*/G.$ The Lie group
$G^+$ is called a
weak quadratic overgroup for
$G.$
Ключевые слова:
coadjoint orbit, quadratic overgroup, weak quadratic overgroup.
Поступило: 05.09.2016
Язык публикации: английский
