Affine structures on a class of virtually nilpotent groups

Abstract

AbstractIn 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.In the 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.This criterium 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.Finally, our results are also translated into the language of complementary submodules in the universal enveloping algebra of a nilpotent Lie algebra