arXiv subjects

Ariane Luzia dos Santos

Publications and source records attributed to Ariane Luzia dos Santos.

2 recordsLinked to original sources

arXiv2015

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method started in Santos-San Martin \cite{ariasm} to get semigroup generators of semi-simple Lie groups. Consider a subset $Γ\subset G$ that contains a semi-simple subgroup $G_{1}$ of $G$. Then $Γ$ generates $G$ if $\mathrm{Ad}\left( Γ\right) $ generates a Zariski dense subgroup of the algebraic group $\mathrm{Ad}\left( G\right) $. The proof is reduced to check that some specific closed orbits of $G_{1}$ in the flag manifolds of $G$ are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups $G$ and subgroups $G_{1}\subset G$.

math.AT↗

arXiv2011

Controllability of control systems simple Lie groups and the topology of flag manifolds

Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group $G$. It is proved that $S=G$ if $S$ contains a subgroup $G(α) \approx \mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm α}$, where $\mathfrak{g}%_α$ is the root space of the root $α$. The proof uses the fact, proved before, that the invariant control set of $S$ is contractible in some flag manifold if $S$ is proper, and exploits the fact that several orbits of $G(α)$ are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.

math.OC↗