Sous-algèbres de Cartan des algèbres de Kac-Moody réelles presque déployées

Abstract

The classification of almost split real forms of symmetrizable Kac-Moody Lie algebras is a rather straightforward infinite-dimensional generalization of the classification of real semi-simple Lie algebras in terms of the Tits index [J. Algebra, 171, 43--96 (1995)]. We study here the conjugate classes of their Cartan subalgebras under the adjoint groups or the full automorphism groups. Maximally split Cartan subalgebras of an almost split real Kac-Moody Lie algebra are mutually conjugate and one can generalize the Sugiura classification (given for real semi-simple Lie algebras) by comparing any Cartan subalgebra to a standard maximally split one. As in the classical case, we prove that the number of conjugate classes of Cartan subalgebras is always finite.