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

H. Ben Messaoud ; M. Layouni

Journal of Lie theory, Tome 32 (2022) no. 4, pp. 937-971

Résumé

We are interested in the gradations of symmetrizable Kac-Moody Lie algebras $\mathfrak g$ 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 $\mathfrak g$ is decomposable, then any indecomposable component of $\mathfrak g$ 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 $\mathfrak g$ admitting adapted root bases. Namely, it is shown that, for such a gradation, there exists a regular standard Kac-Moody-subalgebra $\mathfrak g(I_{re})$ of $\mathfrak g$ containing the grading Kac-Moody Lie subalgebra $\mathfrak m$ 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 $\mathfrak m$ in comparison with those of the graded Kac-Moody Lie algebra $\mathfrak g$ 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 $\mathfrak g$.

Détail

Classification : 17B67
Mots-clés : Kac-Moody Lie algebra, gradation by a Kac-Moody root system, C-admissible pair