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

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 afiEme 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(rho). For an admissible quotient rho : I -> (I) over bar we build also a Kac-Moody subalgebra le which grades finitely the Lie algebra 9. 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.