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

Abstract

For a perfect field F F of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over F F contains an ad-nilpotent element. (2) There are no simple Lie algebras over F 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 ( C 1 ) ({C_1}) of arbitrary characteristic (in particular, over any finite field). Furthermore, we give a sufficient condition to 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 ) {\text {sl}}(2,F) are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field F F is perfect with char ( F ) ≠ 2 {\text {char}}(F) \ne 2 . All Lie algebras considered are finite dimensional.