Characteristically nilpotent Lie algebras: a survey

Abstract

Jacobson proved in 1955 that any Lie algebra over a field of characteristic zero which has nondegenerate derivations is nilpotent. Dixmier and Lister proved in 1957 that the converse is false. They provided an example of a new class of Lie algebras, the characteristically nilpotent Lie algebras (CNLA). Initially it was believed that the class of CLNA was scarce. In 1988 Khakimdzhanov proved that almost all deformations of the nilradical of Borel subalgebras of complex simple Lie algebras are CNLA. That is, CNLA are in abundance within the variety of nilpotent Lie algebras (NLA). Goze and Khakimdzhanov proved in 1994 that for any dimension n 9 an irreducible component of the filiform variety contains an open set consisting of NLA. Vergne proved in 1970 that there are only two naturally graded filiform Lie algebras for even dimensions, Ln and Qn, and only Ln for odd dimensions. Studying the deformations of Ln Khakimdzhanov constructed in 1991 many families of CNLA. In this paper the authors offer a complete survey of the results known thus far about CNLA and add some results which are characteristic of Lie algebras of type Q (NLA which structurally “look like Qn”). They obtain certain results about the rigidity of a NLA and about affine structures over Lie algebras, too. These results are very interesting for cohomology theory and for representation theory of Lie algebras.