Non-singular derivations of solvable Lie algebras in prime characteristic

Abstract

We study solvable Lie algebras in prime characteristic $p$ that admit non-singular derivations. We show that Jacobson's Theorem remains true if the quotients of the derived series have dimension less than~$p$. We also study the structure of Lie algebras with non-singular derivations in which the derived subalgebra is abelian and has codimension~one. The paper presents some new examples of solvable, but not nilpotent, Lie algebras of derived length~3 with non-singular derivations.