Abstract
Kashiwara's construction of the crystal basis for simple integrable modules of U q( g) involves independent sets of axioms, each corresponding to a single simple root, hence to a U q( sl2) case. It seems promising to extend this theory to the case of a U q( g) Verma module. Now such a module, seen as a module for the subalgebra of U q( g) generated by the elements corresponding to a simple root, is a direct sum of U q( 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.
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