Finite-dimensional Lie algebras with a nonsingular derivation

Periodic derivations and prederivations of Lie algebras

“…The only result on periodic derivations in characteristic zero we could find is the following proposition of [7]. Proposition 2.…”

“…We always assume that g is finite-dimensional and complex, if not mentioned otherwise. Denote by Der(g) the Lie algebra of derivations of g. Periodic derivations have been defined in [7]. Our aim is to characterize complex Lie algebras admitting a periodic derivation.…”

“…Proof. There is a proof given in [7] using a binomial formula and determinants, which works for arbitrary characteristic. For the complex numbers there is a shorter argument available.…”

“…It says that such Lie algebras are abelian, if the order of the periodic derivation is not a multiple of six. There is nothing said in [7] on Lie algebras admitting a periodic derivation of order which is a multiple of six. The aim of this paper is to close this gap by characterizing such Lie algebras.…”

Graded Lie algebras of maximal class, III

“…Kostrikin, and M.I. Kuznetsov in their papers [2,13] determined all simple finite-dimensional Lie algebras (over fields of characteristic bigger than seven) admitting non-singular derivations and proved that the existence of any non-singular derivation implies the existence of a non-singular derivation which preserves a cyclic grading. In particular, the Lie algebras preserving a cyclic grading with onedimensional component were proved to be Hamiltonian, coinciding with those referred to as Albert-Frank algebras in the papers of this series.…”