Generators of simple Lie algebras II

TL;DR

This paper investigates the generation properties of simple Lie algebras, showing that certain Cartan type algebras do not possess the 'one-and-a-half generation' property, contrasting with classical Lie algebras.

Contribution

It extends previous work by analyzing the generation properties of Cartan type simple Lie algebras in arbitrary characteristic.

Findings

Cartan type S, H, K Lie algebras lack the 'one-and-a-half' generation property

Classical Lie algebras can have this property, unlike Cartan types

Study of centralisers in Cartan type Lie algebras underpins the results

Abstract

This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic (see arXiv:0708.1711). We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S,H or K never enjoy the 'one-and-a-half generation' property. The methods rely on a study of centralisers in Cartan type Lie algebras.