Geodesic
Cartan Decompositions and Semigroups of Simple Lie Groups
Journal of Lie theory, Tome 28 (2018) no. 1, pp. 187-21
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Let G be a split real connected simple Lie group and S a semigroup of G that contains a subgroup G(α) for an arbitrary root α, isomorphic to SL(2,R). We present a Cartan decomposition of the Lie algebra of G, related to α, invariant by the adjoint action of the Lie algebra sl(2,R) that allows to characterize some properties of the Lie saturate of the semigroup S. We give necessary and sufficient conditions for S to be equal to the whole group G.
Classification : 22E46, 17B22, 93B05
Mots-clés : Semi-simple Lie groups, root systems, controllability 
@article{JLT_2018_28_1_JLT_2018_28_1_a9,
     author = {R. El Assoudi-Baikari},
     title = {Cartan {Decompositions} and {Semigroups} of {Simple} {Lie} {Groups}},
     journal = {Journal of Lie theory},
     pages = {187--21},
     year = {2018},
     volume = {28},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2018_28_1_JLT_2018_28_1_a9/}
}
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R. El Assoudi-Baikari. Cartan Decompositions and Semigroups of Simple Lie Groups. Journal of Lie theory, Tome 28 (2018) no. 1, pp. 187-21. http://geodesic.mathdoc.fr/item/JLT_2018_28_1_JLT_2018_28_1_a9/