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 to get semigroup generators of semi-simple Lie groups. Consider a subset Γ ⊂ G that contains a semi-simple subgroup G1 of G. If one can show that Γ does not leave invariant a contractible subset on any flag manifold of G, then Γ generates G if Ad (Γ) generates a Zariski dense subgroup of the algebraic group Ad (G). The proof is reduced to check that some specific closed orbits of G1 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 G1⊂ G.