
Journal of Lie Theory 28 (2018), No. 1, 001031 Copyright Heldermann Verlag 2018 Cartan Pairs and Shared Orbit Pairs Boris Sirola Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia sirola@math.hr [Abstractpdf] 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 BrylinskiKostant 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 [ Fulltextpdf (396 KB)] for subscribers only. 