
Journal of Lie Theory 24 (2014), No. 2, 321350 Copyright Heldermann Verlag 2014 KacMoody Lie Algebras Graded by KacMoody Root Systems Hechmi Ben Messaoud Université de Monastir, Faculté des Sciences, Dép. de Mathématiques, 5019 Monastir, Tunisia hechmi.benmessaoud@fsm.rnu.tn Guy Rousseau Université de Lorraine, CNRS, UMR 7502, Institut Elie Cartan, 54506 VandoeuvrelesNancy, France guy.rousseau@univlorraine.fr [Abstractpdf] \def\g{{\frak g}} We look to gradations of KacMoody Lie algebras by KacMoody root systems with finite dimensional weight spaces. We extend, to general KacMoody Lie algebras, the notion of $C$admissible pair as introduced by H. Rubenthaler and J. Nervi for semisimple and affine Lie algebras. If $\g$ is a KacMoody Lie algebra (with Dynkin diagram indexed by $I$) and $(I,J)$ is such a $C$admissible pair, we construct a $C$admissible subalgebra $\g^J$, which is a KacMoody Lie algebra of the same type as $\g$, and whose root system $\Sigma$ grades finitely the Lie algebra $\g$. For an admissible quotient $\rho: I\to\overline I$ we build also a KacMoody subalgebra $\g^\rho$ which grades finitely the Lie algebra $\g$. If $\g$ is affine or hyperbolic, we prove that the classification of the gradations of $\g$ is equivalent to those of the $C$admissible pairs and of the admissible quotients. For general KacMoody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized $C$admissible pairs. Keywords: KacMoody algebra, Cadmissible pair, gradation. MSC: 17B67 [ Fulltextpdf (503 KB)] for subscribers only. 