A Note on Automorphisms and Derivations of Lie Algebras

Abstract

In a recent paper Borei and Serre proved the theorem: If $\mathfrak{L}$ is a Lie algebra of characteristic 0 and $\mathfrak{L}$ has an automorphism of prime period without fixed points ≠ 0, then $\mathfrak{L}$ is nilpotent.¹ In this note we give a proof valid also for characteristic p ≠ 0. By the same method we can prove several other similar results on automorphisms and derivations. Our method is based on decompositions of the Lie algebra which determine weakly closed sets of linear transformations. Such a set $\mathfrak{W}$ has, by definition, the closure property that if $A,\,B \in \mathfrak{W}$ then there exists a γ(A, B) in the base field such that $A\,B + \gamma BA \in \mathfrak{W}$ The main result we shall need is the generalized Engel theorem that if $\mathfrak{W}$ is a weakly closed set of nilpotent linear transformations in a finite-dimensional vector space, then the enveloping associative algebra $\mathfrak{W}*$ of $\mathfrak{W}$ is nilpotent [3].