Non-singular derivations of solvable Lie algebras in prime characteristic

TL;DR

This paper investigates solvable Lie algebras over fields of prime characteristic that admit non-singular derivations, extending Jacobson's Theorem under certain conditions and exploring their structure with new examples.

Contribution

It extends Jacobson's Theorem to specific solvable Lie algebras in prime characteristic and provides new examples of non-nilpotent solvable Lie algebras with non-singular derivations.

Findings

Jacobson's Theorem holds if derived series quotients have dimension less than p

Characterization of Lie algebras with abelian derived subalgebra and codimension one

New examples of solvable Lie algebras of derived length 3 with non-singular derivations

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.