Lattices in S-adic Lie Groups

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 grades finitely the Lie algebra g.For an admissible quotient : I I we build also a Kac-Moody subalgebra g 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.