Faithful irreducible representations of modular Lie algebras

Abstract

Let L be a finite-dimensional Lie algebra over a field of characteristic p = 0. By a theorem of Jacobson, L has a finite-dimensional faithful completely reducible module. We show that if F 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.