Faithful completely reducible representations of modular Lie algebras

Abstract

Abstract. Let L be a Lie algebra of dimension n over a field F of characteristic p > 0. I prove the existence of a faithful completely reducible L-module of dimension less than or equal to p n 2 −1 .

Faithful irreducible representations of modular Lie algebras

“…Thus without loss of generality, we may assume that L is restricted. Further, by [3,Corollary 2.3], we may assume that the p-operation [p] vanishes on Asoc(L), the abelian socle of L. We note that, by the Artin-Schreier Theorem 1 , (see [7,Theorem 11.14] or [4, Theorem 3.1],) our assumption that F is not algebraically closed implies that the algebraic closureF has infinite dimension over F .…”