Affine structures on a class of virtually nilpotent groups

In this paper we study the problem of lifting an affine structure on a given nilpotent group N to a group E containing N as a subgroup of finite index. first part of the paper, we sketch some different, but equivalent, points of view concerning the notion of an affine structure on a nilpotent group. Using these equivalent concepts, we are able to construct, in the second part of the paper, a sufficient condition to have a positive answer to the lifting problem of a given affine structure. riterium enables us to obtain an affine structure on a certain class of virtually 4-step nilpotent groups, the existence of which was not known up till now. y, our results are also translated into the language of complementary submodules in the universal enveloping algebra of a nilpotent Lie algebra.