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Математические заметки, 2017, том 101, выпуск 4, статья опубликована в англоязычной версии журнала
(Mi mzm11368)
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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
Образец цитирования:
L. Abdelmoula, Y. Bouaziz, “Weak Quadratic Overgroups for Type I Solvable Lie Groups of the Form $\mathbb{R}\ltimes\mathbb{R}^{n}$”, Math. Notes, 101:4 (2017), 575–589
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/mzm11368
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