A Note on Derivations of Lie Algebras

Abstract

In this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D ≤ Der(L), with the propertyfor some n > 1, is necessarily solvable. As a result, we show that if2010 Mathematics subject classification: primary 17B40.

A note on solvable and nilpotent Lie algebras

“…In a recent paper Shahryari [2] proved a similar theorem for Lie algebras derivations. So we will write about this theorem.…”

“…After that we see if a finite group has a fixed point free automorphism, it is solvable. By using this result Shahryari [2] proved that a finite dimensional lie algebra of characteristic zero, is solvable if it has an abelian subalgebra A with the property [ , ] n L A L  for some 1 n…”

Existentially closed structures and some embedding theorems

“…In [6], we proved that if a Lie algebra L over a filed of characteristics zero has a finite dimensional ideal U such that U n ⊆ [A, U ] for some abelian subalgebra A and n ≥ 2, then U is solvable. Here [A, U ] is the linear span of the set {[a, x] : a ∈ A, x ∈ U }.…”

Character Clusters for Lie Algebra Modules Over a Field of Nonzero Characteristic

“…Lie algebras are very important and have been actively investigated by many authors. See [3][4][5][6] for examples of recent results.…”