Simple Lie algebras of small characteristic VI. Completion of the classification

TL;DR

This paper completes the classification of finite-dimensional simple Lie algebras over algebraically closed fields of characteristic p>3, showing they are of classical, Cartan, or Melikian type, with a focus on the role of nonstandard tori.

Contribution

It proves that if the p-envelope contains nonstandard tori of maximal dimension, then p=5 and the algebra is Melikian, completing the classification.

Findings

Any finite-dimensional simple Lie algebra over F is classical, Cartan, or Melikian.

Nonstandard tori of maximal dimension occur only when p=5, corresponding to Melikian algebras.

The classification is now complete for characteristic p>3.

Abstract

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of F characteristic p>3. We prove that if the p-envelope of L in the derivation algebra of L contains nonstandard tori of maximal dimension, then p=5 and L is isomorphic to one of the Melikian algebras. Together with our earlier results this implies that any finite-dimensional simple Lie algebra over F is of classical, Cartan or Melikian type.