Generators of simple Lie algebras in arbitrary characteristics

TL;DR

This paper investigates the minimal number of generators needed for simple Lie algebras over fields of characteristic 0 or greater than 3, establishing that two generators suffice and exploring the 'one and a half generation' property.

Contribution

It proves that all simple Lie algebras in the specified characteristics can be generated by two elements and characterizes which algebras have the 'one and a half generation' property.

Findings

Any simple Lie algebra in characteristic 0 or p > 3 can be generated by 2 elements.

Classical simple Lie algebras have the 'one and a half generation' property.

Zassenhaus algebras are the only simple Cartan type algebras of type W with this property.

Abstract

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.