Maximal subalgebras of Cartan type in the exceptional Lie algebras

TL;DR

This paper investigates the structure of maximal subalgebras of exceptional simple Lie algebras over algebraically closed fields of positive characteristic, focusing on non-classical types and identifying the first Witt algebra as a possible maximal subalgebra.

Contribution

It characterizes when the first Witt algebra can be a maximal subalgebra of exceptional Lie algebras in positive characteristic.

Findings

Only the first Witt algebra can occur as a non-classical maximal subalgebra.

Provides explicit conditions for maximality of the Witt algebra in .

Advances understanding of subalgebra structure in exceptional Lie algebras.

Abstract

In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras \g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for \g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of \g and give explicit details on when it may be maximal in \g.