On a class of Lie algebras

In the previous paper [T], we have given an estimate for the dimensionality of the derivation algebra of a Lie algebra L satisfying the condition that (ad x) = 0 for x e L implies ad x = 0. Such a Lie algebra will be referred to as an (A2)-algebra in this paper according to the definition given in Jδichi pΓ|, which investigates the (A^)-algebras, k I> 2, with intention to obtain the analogues to the (A)-algebras. He showed that the (A2)-algebras have a different situation from the other classes of (A^)-algebras, k 2>3. But the problem of characterizing the (A2)-algebras remains unsolved. The purpose of this paper is to make a detailed study of this class of Lie algebras. It is known [3] that every semisimple Lie algebra over the field of complex numbers contains no non-zero element x with (ad x) = 0. We shall show that every Lie algebra over a field Φ of characteristic Φ 2 whose Killing form is non-degenerate has the same property. By making use of this result we shall show that, when the basic field Φ is of characteristic 0, L is an (A2> algebra if and only if every element x of the nil radical iVsuch that (ad#) = 0 belongs to the center Z(L), and if and only if L is either reductive, or L^N^Z(N)=Z(L)^Nφ(0) and (ad #)SM) for any xeN\Z(L). This characterization will be used in classifying certain types of solvable (A2)-algebras. A solvable (A2)-algebra is not generally abelian. We shall show that if Φ is an algebraically closed field of characteristic 0, then every solvable (A2> algebra over a field Φ is abelian. The latter half of the paper will be devoted to the study of solvable (A2)-algebras, in particular, to the study of solvable (A2)-algebras L such that dim N/Z(L) is 2 or 3 and of solvable (A2)-algebras of low dimensionalities.