Characteristically nilpotent Lie algebras

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Jacobson’s theorem says that if a finite-dimensiona complex Lie algebra L has a nondegenerate derivation, then L is nilpotent. The converse to this theorem is false. That is, there are nilpotent Lie algebras all of whose derivations are nilpotent. Such algebras are called characteristically nilpotent. The first example of such an algebra was given by J. Dixmier and W. G. Lister [Proc. Amer. Math. Soc. 8 (1957), 155–158; In this paper, the authors give an overview of what is known about the existence and density of various types of characteristically nilpotent Lie algebras and the main techniques used in their construction. For example, there are no characteristically nilpotent Lie algebras of dimension up to 6 (where there are only finitely many isomorphism classes of nilpotent Lie algebras), whereas they do exist for all dimensions above 6. In higher dimensions, there are infinitely many isomorphism classes and one looks for density results. Consider the variety Nn of nilpotent Lie algebra laws of dimension n. It is known that the set of characteristically nilpotent algebras is not (Zariski) open in Nn for n 8. However, the authors show, by giving explicit examples, that each irreducible component of N8 has an open set of characteristically nilpotent algebras. The most widely studied case is when the algebra is filiform, that is, has maximal nilpotency index. In this case, we have that any irreducible component of the variety Fn of filiform laws contains a Zariski open subset of characteristically nilpotent algebras for all n 8. Much less is known about the situation for non-filiform algebras. The authors summarize current knowledge for some special cases, such as characteristically nilpotent algebras obtained from nilradicals of Borel subalgebras of simple algebras. At the end of the paper, the authors also consider those algebras for which the derivation algebra is itself characteristically nilpotent. They give an example of such an algebra and conjecture a structural condition. They also pose some questions related to generalizations of this situation. Most results, especially known classification results, are stated without proof, the authors instead citing original publications. The 42 references provide a good entry into the field for a reader wishing to dig deeper.
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