Sous-algèbres de Cartan des algèbres de Kac-Moody affines réelles presque compactes.

Hechmi Ben Messaoud (1) , Guy Rousseau (2)

1 Département de Mathématiques [Monastir]
2 IECN - Institut Élie Cartan de Nancy

Almost compact real forms of affine Kac-Moody Lie algebras have been already classified [J. Algebra 267, 443-513]. In the present paper, we study the conjugate classes of their Cartan subalgebras under the adjoint groups or the full automorphism groups. Maximally compact Cartan subalgebras are all conjugated to a standard one $h$ and one may compare any Cartan subalgebra to $h$. Cartan subalgebras are related to non compact unitary roots of $h$ and one can see especially that the number of the conjugate classes is always finite. This approach is a generalization of the classification by J. Carmona of Cartan subalgebras for real semi-simple Lie algebras which is different from (but equivalent to) that of Sugiura. The approach of Sugiura which consists in comparing any Cartan subalgebra to a maximally split one does not adapt to our framework of study, as maximally split Cartan subalgebras are not conjugated in general.

Mots clés: Kac-Moody algebra; Cartan subalgebra; almost compact; real form

Domaines: Théorie des groupes [math.GR]

Hechmi Ben Messaoud, Guy Rousseau. Sous-algèbres de Cartan des algèbres de Kac-Moody affines réelles presque compactes.. Journal of Lie Theory, 2007, 17, pp.1-25. ⟨hal-00094832⟩