Kac-Moody Lie algebras graded by Kac-Moody root systems

Abstract

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 \rightarrow 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.