Characterizing nilpotent Lie algebras rely on the dimension of their $2$-nilpotent multipliers

TL;DR

This paper investigates the structure of nilpotent Lie algebras through their 2-nilpotent multipliers, providing characterizations for cases where the invariant s_2(L) ranges from 0 to 6 and identifying 2-capability.

Contribution

It extends previous characterizations of nilpotent Lie algebras by analyzing the structure for s_2(L) between 0 and 6 and determines which are 2-capable.

Findings

Dimension formula for the 2-nilpotent multiplier.

Complete characterization for s_2(L)=0.

Identification of 2-capable algebras within the studied range.

Abstract

There are some results on nilpotent Lie algebras $ L $ investigate the structure of $ L $ rely on the study of its $2$-nilpotent multiplier. It is showed that the dimension of the $2$-nilpotent multiplier of $ L $ is equal to $ \frac{1}{3} n(n-2)(n-1)+3-s_2(L).$ Characterizing the structure of all nilpotent Lie algebras has been obtained for the case $ s_2(L)=0.$ This paper is devoted to the characterization of all nilpotent Lie algebras when $ 0\leq s_2(L)\leq 6.$ Moreover, we show that which of them are $2$-capable.