On certain properties of Lie algebras

Introduction. I. M. Singer [V] has introduced the following condition for a Lie algebra L: (A) Any pair of elements x, y of L such that [_χ, Qx, yHH = O satisfies \^x, y~3 = 0. M. Sugiura [2Γ\ called a Lie algebra satisfying this condition to be an (A)-algebra and proved, among other results, that a Lie algebra L over a field of characteristic 0 is an (A)-algebra if and only if any xeL such that (ad xf = 0 for some k 2> 2 satisfies ad x = 0. On the other hand, S. Togo [3] has considered a Lie algebra L satisfying the condition that (ad^) = 0 implies ad χ = 0, and has given an example of such a Lie algebra which is solvable but not abelian. This is not an (A)-algebra since any solvable (A)-algebra is abelian OH, Ĉ ID Thus we are led to consider a Lie algebra which satisfies the condition that (ad^)^^O implies ad:χ; = 0 for a fixed integer k 2>2. We shall call such a Lie algebra to be an (A^)-algebra. In this paper we shall investigete the properties of (A^)-algebras. It will be shown that a solvable (resp. nilpotent) Lie algebra over a field of characteristic 0 is an (A^)-algebra with k J>3 (resp. k |>2) if and only if it is abelian. We shall show that an (A2)-algebra is not always an (A^)-algebra with k J> 3, much less an (A)-algebra. As to (A^-algebras with k I> 3, if the basic field is algebraically closed and of characteristic 0, we can show that an (A^)-algebra is abelian and so an (A)-algebra. A detailed discussion about (A2)-algebras is also given. The author wishes to express his gratitude to Dr. S. Togo for his encouragement given during the preparation of this paper.