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

Abstract:

Let $(\mathfrak g,\mathfrak g_1)$ be a pair of Lie algebras, defined over a field of characteristic zero, where $\mathfrak g$ is semisimple and $\mathfrak g_1$ is a subalgebra reductive in $\mathfrak g$. We prove a result giving a necessary and sufficient technical condition so that the following holds: ($\boldsymbol {\mathsf {Q}1}$) For any Cartan subalgebra $\mathfrak h_1\subseteq \mathfrak g_1$ there exists a unique Cartan subalgebra $\mathfrak h\subseteq \mathfrak g$ containing $\mathfrak h_1$. Next we study a class of pairs $(\mathfrak g,\mathfrak g_1)$, satisfying ($\boldsymbol {\mathsf {Q}1}$), 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.