Non-Nilpotent Lie Algebras with Non-Singular Derivations

Abstract

Let L be a Lie algebra and (..) be a derivation of L. The derivation (..) is non-singular if it is injective as linear transformation. By a well-known result of N. Jacobson, a Lie algebra of finite dimension over a field of characteristic zero having a non-singular derivation is nilpotent.Although we know that this result is not valid in characteristic p>0, little is known about Lie algebras in p characteristic with non-singular derivations. In this text, we explore the structure of solvable, non-nilpotent Lie algebras with non-singular derivations. We present a new concept for derivations, called Compatible Pairs. This concept is used, for example, to calculate the derivations of an extension of Lie algebras. Another application obtained was a version of Jacobsons Theorem for Lie algebras over fields characteristic p>0. Using Compatible pairs it was possible to obtain a characterization of non-nilpotent Lie algebras, with an abelian deal of codimension 1 and non-singular derivations. Further, a new example of non-nilpotent Lie algebras, with non-singular derivations and arbitrarily nilpotency class was constructed. Finally, we prove that if H is the Heisenberg algebra over a field of characteristic p>0, and I is aH-module such that the semi-direct sum of H and I, is a non-nilpotent Lie algebras with nonsingular derivation, then the dimension of I is, at least, p + 3.