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Journal of Lie Theory 28 (2018), No. 1, 187--210
Copyright Heldermann Verlag 2018

Cartan Decompositions and Semigroups of Simple Lie Groups

Rachida El Assoudi-Baikari
Laboratoire de Mathématiques, INSA de Rouen Normandie, 76801 Saint-Etienne-du-Rouvray, France

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.

Keywords: Semi-simple Lie groups, root systems, controllability.

MSC: 22E46, 17B22, 93B05

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