*Gradings on Lie algebras and endomorphisms of nilpotent groups*

Abstract

For a Lie algebra, it is natural to study the existence of various
gradings: for instance, a Lie algebra is called contractable if it
admits a grading in positive integers, and called Carnot if moreover
such a grading can be generated in degree one. For finite-dimensional
Lie algebra, the existence of such gradings is invariant under taking
field extensions. Besides, given a finitely generated torsion-free
nilpotent group, we characterize several of its properties in terms of
the existence of gradings on the Malcev Lie algebra. For instance, we
show that its systolic growth is equivalent to the growth if and only
if the Lie algebra is Carnot.