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

TL;DR

This paper develops a topological method to identify generators of semi-simple Lie groups by analyzing the topology of their flag manifolds, extending previous work with new cases.

Contribution

It generalizes the topological approach to find generators of semi-simple Lie groups, considering multiple cases and using algebraic topology of flag manifolds.

Findings

Generators can be identified via Zariski dense subgroups of the adjoint representation.

Specific closed orbits in flag manifolds are non-trivial in algebraic topology.

The method applies to various semi-simple Lie groups and subgroups.

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 $G_{1}\subset G$.