Cartan subspaces of symmetric Lie algebras

Abstract:

A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra $\mathfrak {g}$ with a decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {p}$ such that $\mathfrak {k}$ is a subalgebra of $\mathfrak {g},[\mathfrak {k},\mathfrak {p}] \subset \mathfrak {p}$ and $[\mathfrak {p},\mathfrak {p}] \subset \mathfrak {k}$. A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.