On Kirillov’s lemma and the Benson–Ratcliff invariant

Abstract

In this paper, we study the conjecture of Benson and Ratcliff, which deals with the class of nilpotent Lie algebras of a one-dimensional center. We show that this conjecture is true for any nilpotent Lie algebra g with dimg≤5, but it fails for the dimensions greater or equal to 6. To this end, we produce counter-examples to the Benson–Ratcliff conjecture in all dimensions n ≥ 6. Finally, we show that this conjecture is true for the class of three-step nilpotent Lie algebras and for some other classes of nilpotent Lie algebras.