Faithful irreducible representations of modular Lie algebras

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.