On Cartan subalgebras and Cartan subspaces of nonsymmetric pairs of Lie algebras

Abstract

Let ( g , g 1 ) (\mathfrak g,\mathfrak g_1) be a pair of Lie algebras, defined over a field of characteristic zero, where g \mathfrak g is semisimple and g 1 \mathfrak g_1 is a subalgebra reductive in g \mathfrak g . We prove a result giving a necessary and sufficient technical condition so that the following holds: ( Q 1 \boldsymbol {\mathsf {Q}1} ) For any Cartan subalgebra h 1 ⊆ g 1 \mathfrak h_1\subseteq \mathfrak g_1 there exists a unique Cartan subalgebra h ⊆ g \mathfrak h\subseteq \mathfrak g containing h 1 \mathfrak h_1 . Next we study a class of pairs ( g , g 1 ) (\mathfrak g,\mathfrak g_1) , satisfying ( Q 1 \boldsymbol {\mathsf {Q}1} ), which we call Cartan pairs. For such pairs and the corresponding Cartan subspaces, we prove some useful results that are classical for symmetric pairs. Thus we extend a part of the previous research on Cartan subspaces done by Dixmier, Lepowsky and McCollum.