On Crystal Bases and Enright's Completions

We investigate the interplay of crystal bases and completions in the sense of Enright on certain nonintegrable representations of quantum groups. We define completions of crystal bases, show that this notion of completion is compatible with Enright's completion of modules, prove that every module in our category has a crystal basis which can be completed and that a completion of the crystal lattice is unique. Furthermore, we give two constructions of the completion of a crystal lattice.