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

Abstract

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 $\Gamma \subset G$ that contains a semi-simple subgroup $G_1$ of $G$. Then $\Gamma$ generates $G$ if $\mathrm{Ad}\left(\Gamma \right)$ 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 $G_1\subset G$.