A Note on Lie Algebras of Characteristic p

Abstract

Let $\mathfrak{L}$ be a finite dimensional Lie algebra over a field of characteristic p, let $\mathfrak{A}$ be the universal associative algebra of $\mathfrak{L}$ ([1] and [4]) and let $\mathfrak{C}$ be the center of $\mathfrak{A}$. In this note we prove that if a is a linear element of $\mathfrak{A}$ then there exists a non-zero polynomial φ such that <j> (a) ε $\mathfrak{C}$. We use this result to obtain the following: (1) a simple direct proof of Iwasawa’s theorem ([2], p. 420) that every finite dimensional Lie algebra of characteristic p has a faithful finite dimensional representation, (2) a proof of a conjecture of Chevalley that every finite dimensional Lie algebra of characteristic p has a representation which is not completely reducible, (3) a proof that $\mathfrak{A}$ can be imbedded in a division algebra.