Translations in simply transitive affine actions of 5-dimensional nilpotent Lie groups

In this paper we determine all 5-dimensional nilpotent Lie groups acting simply transitively and affinely on R5 in such a way that only the identity element acts as a pure translation. In fact, a complete classification of all possible such actions is obtained by translating the original problem in terms of complete left symmetric structures on the corresponding Lie algebra. It turns out that some 5-dimensional nilpotent Lie groups admit no such actions, while others admit a finite number of such actions, and still others even admit infinite families of such actions.