Characterizing nilpotent Lie algebras that satisfy on converse of the Schur's theorem

Abstract

Let $ L $ be a finite dimensional nilpotent Lie algebra and $ d $ be the minimal number generators for $ L/Z(L). $ It is known that $ \dim L/Z(L)=d \dim L^{2}-t(L)$ for an integer $ t(L)\geq 0. $ In this paper, we classify all finite dimensional nilpotent Lie algebras $ L $ when $ t(L)\in \lbrace 0, 1, 2 \rbrace.$ We find also a construction, which shows that there exist Lie algebras of arbitrary $ t(L). $