Gradings on Lie algebras and endomorphisms of nilpotent groups
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.