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.
Publié le : 2007-07-05
Classification:
Kac-Moody algebra,
Cartan subalgebra,
almost compact,
real form,
17B67,
[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]
@article{hal-00094832,
author = {Ben Messaoud, Hechmi and Rousseau, Guy},
title = {Sous-alg\`ebres de Cartan des alg\`ebres de Kac-Moody affines r\'eelles presque compactes.},
journal = {HAL},
volume = {2007},
number = {0},
year = {2007},
language = {fr},
url = {http://dml.mathdoc.fr/item/hal-00094832}
}
Ben Messaoud, Hechmi; Rousseau, Guy. Sous-algèbres de Cartan des algèbres de Kac-Moody affines réelles presque compactes.. HAL, Tome 2007 (2007) no. 0, . http://gdmltest.u-ga.fr/item/hal-00094832/