Periodic derivations and prederivations of Lie algebras

TL;DR

This paper characterizes finite-dimensional complex Lie algebras with periodic derivations, showing they are at most two-step nilpotent and exploring related gradings and derivation properties.

Contribution

It provides new characterizations of such Lie algebras, including gradings by roots of unity and properties of inverse derivations, expanding understanding of their structure.

Findings

Lie algebras with periodic derivations are at most two-step nilpotent

Existence of gradings by sixth roots of unity characterizes these algebras

Results on periodic prederivations and generalizations of Engel-4-Lie algebras

Abstract

We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several characterizations, such as the existence of gradings by sixth roots of unity, or the existence of a nonsingular derivation whose inverse is again a derivation. We also obtain results on the existence of periodic prederivations. In this context we study a generalization of Engel-4-Lie algebras.