Faithful irreducible representations of modular Lie algebras

TL;DR

This paper investigates conditions under which finite-dimensional Lie algebras over fields of non-zero characteristic have faithful irreducible modules, extending Jacobson's theorem to non-algebraically closed fields and providing necessary and sufficient criteria.

Contribution

It extends Jacobson's theorem by establishing the existence of faithful irreducible modules over non-algebraically closed fields and characterizes when such modules exist over algebraically closed fields.

Findings

Existence of faithful irreducible modules over non-algebraically closed fields.

Necessary and sufficient conditions for faithful irreducible modules over algebraically closed fields.

Extension of Jacobson's theorem to broader field settings.

Abstract

Let L be a finite-dimensional Lie algebra over a field of non-zero characteristic. By a theorem of Jacobson, L has a finite-dimensional faithful module which is completely reducible. We show that if the field is not algebraically closed, then L has an irreducible such module. We also give a necessary and sufficient condition for a finite-dimensional Lie algebra over an algebraically closed field of non-zero characteristic to have a faithful irreducible module.