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

Varea, V. R.

doi:10.1090/S0002-9939-1988-0962799-X

Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one
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by V. R. Varea PDF

Proc. Amer. Math. Soc. 104 (1988), 363-368 Request permission

Abstract:

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 $({C_1})$ 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 ${\text {sl}}(2,F)$ are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field $F$ is perfect with ${\text {char}}(F) \ne 2$. All Lie algebras considered are finite dimensional.

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