Gradings on Lie algebras with applications to infra-nilmanifolds

TL;DR

This paper investigates gradings on finite-dimensional Lie algebras, proving invariance under field extensions and automorphisms, and applies these results to infra-nilmanifolds to derive new geometric and dynamical properties.

Contribution

It provides new proofs of invariance of gradings under field extensions and automorphisms, and applies these to study properties of infra-nilmanifolds.

Findings

Existence of certain gradings is invariant under field extensions.

Infra-nilmanifold properties depend only on the covering Lie group.

Constructed a nilmanifold with Anosov diffeomorphism but no non-trivial self-covers.

Abstract

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type.