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Journal of Lie Theory 28 (2018), No. 1, 001--031
Copyright Heldermann Verlag 2018

Cartan Pairs and Shared Orbit Pairs

Boris Sirola
Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia


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.

Keywords: Semisimple Lie algebra, Cartan subalgebra, nonsymmetric pair, Kostant pair, Cartan subspace, Cartan pair, restricted root, set of restricted roots, shared orbit pair.

MSC: 17B20; 17B05, 17B22

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