Geodesic
Cartan Pairs and Shared Orbit Pairs
Journal of Lie theory, Tome 28 (2018) no. 1, pp. 1-31
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We study a class of pairs of Lie algebras $(\g,\g_1)$ that we call Cartan pairs; here $\g$ is semisimple and $\g_1$ is a reductive in $\g$ subalgebra. For these pairs, which generalize symmetric ones, we have standardly defined Cartan subspaces, and consequently the set of restricted roots $\Sigma(\g,\a)$. We prove that there are infinitely many interesting nonsymmetric Cartan pairs. Next we prove that every pair of the well known Brylinski-Kostant list of shared orbit pairs is a Cartan pair. As a continuation of the previous research we obtained some further useful and clarifying results and examples related to Cartan pairs and Cartan subspaces.
Classification : 17B20, 17B05, 17B22
Mots-clés : Semisimple Lie algebra, Cartan subalgebra, nonsymmetric pair, Kostant pair, Cartan subspace, Cartan pair, restricted root, set of restricted roots, shared orbit pair 
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     author = {B. Sirola},
     title = {Cartan {Pairs} and {Shared} {Orbit} {Pairs}},
     journal = {Journal of Lie theory},
     pages = {1--31},
     year = {2018},
     volume = {28},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2018_28_1_JLT_2018_28_1_a0/}
}
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B. Sirola. Cartan Pairs and Shared Orbit Pairs. Journal of Lie theory, Tome 28 (2018) no. 1, pp. 1-31. http://geodesic.mathdoc.fr/item/JLT_2018_28_1_JLT_2018_28_1_a0/