Lie algebras whose derivation algebras are simple

TL;DR

This paper characterizes Lie algebras with simple derivation algebras across arbitrary dimensions and classifies those with complete simple derivation algebras over specific fields and gradings.

Contribution

It extends the classification of Lie algebras with simple derivation algebras to arbitrary dimensions and various field characteristics.

Findings

Characterization of Lie algebras with simple derivation algebras

Classification of Lie algebras with complete simple derivation algebras

Results over fields of prime characteristic and zero with specific gradings

Abstract

It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that have a simple derivation algebra. As an application we classify the Lie algebras that have a complete simple derivation algebra and are either finite-dimensional over an algebraically closed field of prime characteristic $p>3$ or $\mathbb{Z}$-graded of finite growth over an algebraically closed field of characteristic zero.