Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one

V. R. Varea
1988 Proceedings of the American Mathematical Society  
For a perfect field F of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over F contains an ad-nilpotent element. (2) There are no simple Lie algebras over F having only abelian subalgebras. They are used to guarantee the existence of an ad-nilpotent element in every Lie algebra over a perfect field of type (Ci) of arbitrary characteristic (in particular, over any finite field). Furthermore, we give a sufficient condition to insure the
more » ... insure the existence of ad-nilpotent elements in a Lie algebra over any perfect field. As a consequence of this result we obtain an easy proof of the fact that the Zassenhaus algebras and sl(2, F) are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field F is perfect with charfF) ^ 2. All Lie algebras considered are finite dimensional. Introduction. It is well known that every Lie algebra over an algebraically closed field contains ad-nilpotent elements. In characteristic zero, it follows from the classical theory. Benkart and Isaacs gave in [3] a proof which works for algebraically closed fields of arbitrary characteristic. In this paper, we prove that for a nonzero nilpotent derivation D of a Lie algebra L, each element of Ker D which acts nilpotently on Ker D is ad-nilpotent on L. We use this result to prove that for a perfect field F the following statements are equivalent: (1) Any Lie algebra over F contains an ad-nilpotent element. (2) There are no simple Lie algebras over F having only abelian subalgebras. Then we show that the following holds: (1) Any Lie algebra over a perfect field of type (Gi) of arbitrary characteristic necessarily contains an ad-nilpotent element. (2) The Brauer group of a perfect field over which every Lie algebra contains ad-nilpotent elements must be trivial (it follows from [6]). Obviously, Lie algebras containing no nonzero element x such that ad x is split cannot contain any ad-nilpotent element. The simplest example of these algebras is the 3-dimensional Lie algebra su(2) of the real vectors with the vector product. We prove that the existence of an element x in each subalgebra S of L such that ad x is split on S guarantees the existence of an ad-nilpotent element in L, provided F is perfect of arbitrary characteristic. Theorem 1.5 below is slightly stronger than this assertion which is obtained from it by taking the subspace V in the statement of the theorem to be zero.
doi:10.1090/s0002-9939-1988-0962799-x fatcat:4fnr7wheyvhdlniuzbt2fa2mq4