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Journal of Lie Theory 24 (2014), No. 2, 321--350
Copyright Heldermann Verlag 2014

Kac-Moody Lie Algebras Graded by Kac-Moody Root Systems

Hechmi Ben Messaoud
Université de Monastir, Faculté des Sciences, Dép. de Mathématiques, 5019 Monastir, Tunisia

Guy Rousseau
Université de Lorraine, CNRS, UMR 7502, Institut Elie Cartan, 54506 Vandoeuvre-les-Nancy, France


\def\g{{\frak g}} We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of $C$-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If $\g$ is a Kac-Moody 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 Kac-Moody 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 Kac-Moody 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 Kac-Moody 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: Kac-Moody algebra, C-admissible pair, gradation.

MSC: 17B67

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