On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Abstract

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log₂ p.

Key Words:

• Characteristic ideal
• Derivation
• Derivated length
• Lie algebra

2000 Mathematics Subject Classification:

• Primary 17B30
• Secondary 17B40

ACKNOWLEDGMENT

This article is supported by DFFD, Grant F25.1/095.

Notes

Communicated by I. Shestakov.