
Journal of Lie Theory 32 (2022), No. 4, 937971 Copyright Heldermann Verlag 2022 On Gradations of Decomposable KacMoody Lie Algebras by KacMoody Root Systems Hechmi Ben Messaoud Dept. of Mathematics, Faculty of Sciences, University of Monastir, Tunisia hechmi.benmessaoud@fsm.rnu.tn Marwa Layouni Dept. of Mathematics, Faculty of Sciences, University of Monastir, Tunisia marwa.layouni@fsm.rnu.tn [Abstractpdf] We are interested in the gradations of symmetrizable KacMoody Lie algebras $\mathfrak g$ by root systems $\Sigma$ of KacMoody type. We first show that we can reduce to the case where the grading root system $\Sigma$ is indecomposable. If the graded KacMoody 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 firstnamed 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 KacMoodysubalgebra $\mathfrak g(I_{re})$ of $\mathfrak g$ containing the grading KacMoody 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 KacMoody Lie subalgebra $\mathfrak m$ in comparison with those of the graded KacMoody 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 $\mathfrak g$. Keywords: KacMoody Lie algebra, gradation by a KacMoody root system, Cadmissible pair. MSC: 17B67. [ Fulltextpdf (275 KB)] for subscribers only. 