On Gradations of Decomposable Kac-Moody Lie Algebras by Kac-Moody Root Systems

We are interested in the gradations of symmetrizable Kac-Moody Lie algebras g by root systems sigma of Kac-Moody type. We first show that we can reduce to the case where the grading root system sigma is indecomposable. If the graded Kac-Moody Lie algebra g is decomposable, then any indecomposable component of g is either fictive (and contributes little to the gradation) or effective (and essentially sigma-graded). Based on work by G. Rousseau and the first-named author, we extend most of the results on finite gradations to the gradations of g admitting adapted root bases. Namely, it is shown that, for such a gradation, there exists a regular standard Kac-Moody-subalgebra g(I-re) of g containing the grading Kac-Moody Lie subalgebra m and which is finitely really sigma-graded. This enables us to investigate the structure of the Weyl group and the Tits cone of the grading Kac-Moody Lie subalgebra m in comparison with those of the graded Kac-Moody Lie algebra g and to prove a conjugacy theorem on adapted pairs of root bases. We end the paper by providing a unified construction for the finite imaginary gradations of g.