Lie algebras with self-centralizing ad-nilpotent elements

Abstract

Let L be a finite dimensional Lie algebra over an arbitrary field F. For x EL, the centraZizer C(x) is the subalgebra {y EL ( [yx] = 0). We say that x is selfcentralizing if dim C(x) = 1, ( i.e. C(x) = Fx). If L contains an ad-nilpotent self-centralizing element, then the structure of L is severely constrained. In Section 2 we shall see that if char(F) = 0, then dim L 0, then examples exist with dimension exceeding 3. It turns out for p > 2 that these algebras are necessarily simple and have p-power dimension. (The situation when p = 2 is more complicated and we do not discuss it.) Let char(F) = p > 2. We say that a polynomial f E F[Xj is a p-polynomial if the only powers of X having nonzero coefficients in f are of the form XP” for i > 0. Given a positive integer n and a p-polynomial f, we construct in Section 4 a specific Lie algebra La(f) of dimension p %. These algebras contain self-centralizing ad-nilpotent elements and for perfect fields L, we show in Section 7 that the L,(f) are the only algebras of dimension >3 which do. The definition of L,(f) is motivated by the special case f = 0 where the algebra can be identified with the Zassenhaus algebra of dimension pn (see Ree [S]). In Section 3, we study the structure of Zassenhaus algebras from the point of view of self-centralizing ad-nilpotent elements. A consequence of this investigation is an easy proof that when F is algebraically closed, the Zassenhaus algebras and o((2, F) are the only finite dimensional simple algebras which have subalgebras of codimension 1. (This was asserted by Amayo [2] without any assumption on F. \Ve give a counterexample in Section 6 which shows that the result is not true over arbitrary fields.) Another generalization of the Zassenhaus algebras was given by Albert and Frank [l]. In Section 5 we show that these “Alberttzassenhaus” algebras over perfect fields also contain self-centralizing ad-nilpotent elements and hence are