Structures cristallines pour une classe d'indécomposables de Uq(sl2)

AbstractKashiwara's construction of the crystal basis for simple integrable modules of Uq(g) involves independent sets of axioms, each corresponding to a single simple root, hence to a Uq(sl2) case. It seems promising to extend this theory to the case of a Uq(g) Verma module. Now such a module, seen as a module for the subalgebra of Uq(g) generated by the elements corresponding to a simple root, is a direct sum of Uq(sl2) indecomposables belonging to a category I. In this article we show there is a theory of crystallisation for I, such that one recovers with some modifications, analogs of the main properties of crystal bases, that is to say: indexation of the basis by a class of oriented graphs admitting tensor products, quasi-orthonormality at q=0 with respect to contravariant forms, and compatibility with left tensorisation by finite modules. We show however that the direct generalisation of the construction of crystal bases to Verma modules fails