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

2015/04/27 by Ariane Luzia dos Santos, Santos, Ariane Luzia dos, Luiz A. B. San Martín +1
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1504.07271

openalex publication_date 2015/04/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

In this paper we continue to develop the topological method started in Santos-San Martin \citeariasm to get semigroup generators of semi-simple Lie groups. Consider a subset Γ⊂ G that contains a semi-simple subgroup G₁ 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.

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