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

Author:
V. R. Varea

Journal:
Proc. Amer. Math. Soc. **104** (1988), 363-368

MSC:
Primary 17B40; Secondary 17B50

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962799-X

MathSciNet review:
962799

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Abstract: For a perfect field of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over contains an ad-nilpotent element. (2) There are no simple Lie algebras over 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 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 are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field is perfect with . All Lie algebras considered are finite dimensional.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0962799-X

Keywords:
Ad-nilpotent element,
modular Lie algebras,
Zassenhaus algebra

Article copyright:
© Copyright 1988
American Mathematical Society