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Mat. Zametki, 2017, Volume 101, Issue 4, paper published in the English version journal (Mi mz11368)  

Papers published in the English version of the journal

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

Abstract: 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 $G.$

Keywords: coadjoint orbit, quadratic overgroup, weak quadratic overgroup.

Funding Agency
This work was completed with the support of D.G.R.S.R.T Research Laboratory LR 11 ES 35.

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English version:
Mathematical Notes, 2017, 101:4, 575–589

Bibliographic databases:

Received: 05.09.2016
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Citation: L. Abdelmoula, Y. Bouaziz, “Weak Quadratic Overgroups for Type I Solvable Lie Groups of the Form <nobr>$\mathbb{R}\ltimes\mathbb{R}^{n}$</nobr>”, Math. Notes, 101:4 (2017), 575–589

Citation in format AMSBIB
\Bibitem{AbdBou17}
\by L.~Abdelmoula, Y.~Bouaziz
\paper Weak Quadratic Overgroups for Type I Solvable Lie Groups of the Form <nobr>$\mathbb{R}\ltimes\mathbb{R}^{n}$</nobr>
\jour Math. Notes
\yr 2017
\vol 101
\issue 4
\pages 575--589
\mathnet{http://mi.mathnet.ru/mz11368}
\crossref{https://doi.org/10.1134/S0001434617030178}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3646053}
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\elib{http://elibrary.ru/item.asp?id=31088149}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85018842559}


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