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

TL;DR

This paper classifies finite dimensional nilpotent Lie algebras based on the parameter t(L), which relates to the structure of the algebra and its center, extending understanding of Lie algebra properties linked to Schur's theorem.

Contribution

The paper provides a complete classification of nilpotent Lie algebras for t(L) values 0, 1, and 2, and introduces a construction method for algebras with arbitrary t(L).

Findings

Classified nilpotent Lie algebras for t(L) in {0, 1, 2}.

Developed a construction method for Lie algebras with any t(L).

Extended the understanding of the structure of nilpotent Lie algebras related to 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). $