A method to find generators of a semi-simple Lie group via the topology of its flag manifolds |
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Authors: | Ariane?Luzia?dos?Santos Email author" target="_blank">Luiz?A?B?San MartinEmail author |
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Institution: | 1.Departamento de Ciências da Educa??o,FCL - Unesp,Araraquara,Brazil;2.Departamento de Matemática,Imecc - Unicamp,Campinas,Brazil |
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Abstract: | In this paper we continue to develop the topological method 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. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold 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 \(G_{1}\subset G\). |
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